86:@0.036727:0.974518:0.057230:0.974518:0.057230:0.956333:0.036727:0.956333:0.000000:0.000000
Límites y continuidad de funciones:@0.073089:0.043661:0.383513:0.043661:0.383513:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Tema 3. Definición precisa de límite de una función:@0.073089:0.098522:0.825373:0.098522:0.825373:0.067281:0.073089:0.067281:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Reto matemático:@0.302768:0.139665:0.425244:0.139665:0.425244:0.124259:0.302768:0.124259:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Si :@0.078581:0.449405:0.093374:0.449405:0.093374:0.435121:0.078581:0.435121:0.000000:0.000000:0.000000
u x y v y z:@0.093287:0.449405:0.197138:0.449405:0.197138:0.434491:0.093287:0.434491:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
= –:@0.102692:0.449935:0.133633:0.449935:0.133633:0.433885:0.102692:0.433885:0.000000:0.000000:0.000000
, :@0.142284:0.449405:0.149798:0.449405:0.149798:0.435121:0.142284:0.435121:0.000000:0.000000
= –:@0.158683:0.449935:0.189641:0.449935:0.189641:0.433885:0.158683:0.433885:0.000000:0.000000:0.000000
, se tiene :@0.198075:0.449405:0.260603:0.449405:0.260603:0.435121:0.198075:0.435121:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
u v x y:@0.078545:0.464534:0.149429:0.464534:0.149429:0.449619:0.078545:0.449619:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
+ = – +( – )= –:@0.087949:0.465064:0.237650:0.465064:0.237650:0.449014:0.087949:0.449014:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y  z:@0.171719:0.464534:0.199280:0.464534:0.199280:0.449619:0.171719:0.449619:0.000000:0.000000:0.000000:0.000000
x z:@0.221569:0.464534:0.245147:0.464534:0.245147:0.449619:0.221569:0.449619:0.000000:0.000000:0.000000
, :@0.246084:0.464534:0.252661:0.464534:0.252661:0.450250:0.246084:0.450250:0.000000:0.000000
y la desigualdad:@0.078495:0.479663:0.185223:0.479663:0.185223:0.465379:0.078495:0.465379:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
|u v| |u| |v|:@0.078495:0.494792:0.182931:0.494792:0.182931:0.479877:0.078495:0.479877:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
+ ≤:@0.092250:0.495322:0.133365:0.495322:0.133365:0.479272:0.092250:0.479272:0.000000:0.000000:0.000000
+:@0.152375:0.495322:0.166314:0.495322:0.166314:0.479272:0.152375:0.479272:0.000000
se transforma en:@0.078495:0.509921:0.188557:0.509921:0.188557:0.495637:0.078495:0.495637:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
|x z| |x y| |y z|:@0.078495:0.525050:0.207145:0.525050:0.207145:0.510135:0.078495:0.510135:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
– ≤ – + –:@0.090560:0.525579:0.195314:0.525579:0.195314:0.509530:0.090560:0.509530:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
; :@0.208098:0.525050:0.214675:0.525050:0.214675:0.510766:0.208098:0.510766:0.000000:0.000000
esto, en términos de :@0.078511:0.540179:0.216673:0.540179:0.216673:0.525895:0.078511:0.525895:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
la distancia entre dos :@0.078511:0.555308:0.221412:0.555308:0.221412:0.541024:0.078511:0.541024:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
números reales, se expresa :@0.078511:0.570437:0.257739:0.570437:0.257739:0.556152:0.078511:0.556152:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
como:@0.078511:0.585566:0.117284:0.585566:0.117284:0.571281:0.078511:0.571281:0.000000:0.000000:0.000000:0.000000
d x, z:@0.078511:0.600695:0.116564:0.600695:0.116564:0.585780:0.078511:0.585780:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.087866:0.601224:0.093438:0.601224:0.093438:0.585175:0.087866:0.585175:0.000000
)≤ (:@0.117535:0.601224:0.153814:0.601224:0.153814:0.585175:0.117535:0.585175:0.000000:0.000000:0.000000:0.000000
d x, y:@0.138887:0.600695:0.177157:0.600695:0.177157:0.585780:0.138887:0.585780:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
)+ (:@0.178128:0.601224:0.214407:0.601224:0.214407:0.585175:0.178128:0.585175:0.000000:0.000000:0.000000:0.000000
d y, z:@0.199480:0.600695:0.236954:0.600695:0.236954:0.585780:0.199480:0.585780:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
):@0.237918:0.601224:0.243490:0.601224:0.243490:0.585175:0.237918:0.585175:0.000000
,:@0.244411:0.600695:0.247323:0.600695:0.247323:0.586410:0.244411:0.586410:0.000000
que hemos obtenido en la :@0.078511:0.615824:0.257323:0.615824:0.257323:0.601539:0.078511:0.601539:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
demostración del punto d. :@0.078511:0.630952:0.258824:0.630952:0.258824:0.616668:0.078511:0.616668:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
del tema 3.2. :@0.078511:0.646081:0.178156:0.646081:0.178156:0.631797:0.078511:0.631797:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Competencia :@0.078511:0.404523:0.177203:0.404523:0.177203:0.389116:0.078511:0.389116:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
matemática :@0.078511:0.419652:0.167125:0.419652:0.167125:0.404245:0.078511:0.404245:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
3.1 Distancia entre dos números reales:@0.293660:0.195804:0.596465:0.195804:0.596465:0.178857:0.293660:0.178857:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
De manera intuitiva, las ideas de proximidad, de tendencia, de aproximación y de :@0.293660:0.223167:0.931067:0.223167:0.931067:0.207454:0.293660:0.207454:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
cercanía a un punto, a un valor numérico o también de tendencia a alejarse están ligadas :@0.293660:0.239809:0.931127:0.239809:0.931127:0.224096:0.293660:0.224096:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
con la distancia entre dos puntos en la recta numérica, la cual definimos a continuación.:@0.293660:0.256451:0.927477:0.256451:0.927477:0.240738:0.293660:0.240738:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Definición. :@0.293660:0.280235:0.379195:0.280235:0.379195:0.263565:0.293660:0.263565:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Sean  , y :@0.379070:0.280235:0.445005:0.280235:0.445005:0.264522:0.379070:0.264522:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.418480:0.280235:0.425935:0.280235:0.425935:0.263829:0.418480:0.263829:0.000000
∈ ℝ:@0.444894:0.280817:0.478027:0.280817:0.478027:0.263163:0.444894:0.263163:0.000000:0.000000:0.000000
. La distancia de   a   se denota :@0.477935:0.280235:0.709278:0.280235:0.709278:0.264522:0.477935:0.264522:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.596312:0.280235:0.628451:0.280235:0.628451:0.263829:0.596312:0.263829:0.000000:0.000000:0.000000
d  x, y:@0.709130:0.280235:0.754486:0.280235:0.754486:0.263829:0.709130:0.263829:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.723782:0.280817:0.729912:0.280817:0.729912:0.263163:0.723782:0.263163:0.000000
):@0.755553:0.280817:0.761683:0.280817:0.761683:0.263163:0.755553:0.263163:0.000000
 y se define como :@0.761591:0.280235:0.893737:0.280235:0.893737:0.264522:0.761591:0.264522:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d ,:@0.556332:0.304019:0.586207:0.304019:0.586207:0.288306:0.556332:0.288306:0.000000:0.000000:0.000000
(:@0.567376:0.304601:0.573506:0.304601:0.573506:0.286947:0.567376:0.286947:0.000000
x  y:@0.574518:0.304019:0.599055:0.304019:0.599055:0.287612:0.574518:0.287612:0.000000:0.000000:0.000000:0.000000
)=∙ – ∙:@0.600104:0.304601:0.659983:0.304601:0.659983:0.286947:0.600104:0.286947:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.628691:0.304019:0.654866:0.304019:0.654866:0.287612:0.628691:0.287612:0.000000:0.000000:0.000000
.:@0.660995:0.304019:0.664198:0.304019:0.664198:0.288306:0.660995:0.288306:0.000000
De la definición de valor absoluto y de la distancia entre dos números reales, se tiene::@0.293642:0.331381:0.909523:0.331381:0.909523:0.315668:0.293642:0.315668:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
  ∙:@0.293660:0.384356:0.329850:0.384356:0.329850:0.336208:0.293660:0.336208:0.000000:0.000000:0.000000
si     y, entonces     y   0, luego, :@0.336027:0.359590:0.587299:0.359590:0.587299:0.343877:0.336027:0.343877:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.350532:0.359590:0.357987:0.359590:0.357987:0.343184:0.350532:0.343184:0.000000
≥:@0.361871:0.360172:0.377204:0.360172:0.377204:0.342518:0.361871:0.342518:0.000000
x:@0.466866:0.359590:0.474321:0.359590:0.474321:0.343184:0.466866:0.343184:0.000000
– ≥:@0.478205:0.360172:0.518572:0.360172:0.518572:0.342518:0.478205:0.342518:0.000000:0.000000:0.000000
d x, y:@0.587157:0.359590:0.629254:0.359590:0.629254:0.343184:0.587157:0.343184:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.597447:0.360172:0.603576:0.360172:0.603576:0.342518:0.597447:0.342518:0.000000
)=∙ – ∙= –:@0.630322:0.360172:0.731801:0.360172:0.731801:0.342518:0.630322:0.342518:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.658909:0.359590:0.686151:0.359590:0.686151:0.343877:0.658909:0.343877:0.000000:0.000000:0.000000
 x y.:@0.708627:0.359590:0.744521:0.359590:0.744521:0.343877:0.708627:0.343877:0.000000:0.000000:0.000000:0.000000:0.000000
si :@0.336045:0.379810:0.350642:0.379810:0.350642:0.364097:0.336045:0.364097:0.000000:0.000000:0.000000
x :@0.350550:0.379810:0.361245:0.379810:0.361245:0.363403:0.350550:0.363403:0.000000:0.000000
< :@0.361190:0.380392:0.381033:0.380392:0.381033:0.362738:0.361190:0.362738:0.000000:0.000000
y, :@0.381033:0.379810:0.394348:0.379810:0.394348:0.363403:0.381033:0.363403:0.000000:0.000000:0.000000
entonces :@0.394194:0.379810:0.465662:0.379810:0.465662:0.364097:0.394194:0.364097:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y   x:@0.465540:0.379810:0.496004:0.379810:0.496004:0.363403:0.465540:0.363403:0.000000:0.000000:0.000000:0.000000:0.000000
–  >:@0.476198:0.380392:0.515884:0.380392:0.515884:0.362738:0.476198:0.362738:0.000000:0.000000:0.000000:0.000000
 :@0.515792:0.379810:0.519105:0.379810:0.519105:0.363403:0.515792:0.363403:0.000000
0, resulta, :@0.518940:0.379810:0.591413:0.379810:0.591413:0.364097:0.518940:0.364097:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x, y:@0.591262:0.379810:0.633359:0.379810:0.633359:0.363403:0.591262:0.363403:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.601552:0.380392:0.607681:0.380392:0.607681:0.362738:0.601552:0.362738:0.000000
)=∙ – ∙:@0.634427:0.380392:0.694306:0.380392:0.694306:0.362738:0.634427:0.362738:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y = y x.:@0.663013:0.379810:0.742220:0.379810:0.742220:0.363403:0.663013:0.363403:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.720186:0.380392:0.729390:0.380392:0.729390:0.362738:0.720186:0.362738:0.000000
        así, :@0.293653:0.413343:0.351323:0.413343:0.351323:0.397630:0.293653:0.397630:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∙:@0.351164:0.429743:0.362309:0.429743:0.362309:0.381595:0.351164:0.381595:0.000000
d x  y:@0.365070:0.404977:0.407039:0.404977:0.407039:0.388570:0.365070:0.388570:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.375360:0.405559:0.381489:0.405559:0.381489:0.387905:0.375360:0.387905:0.000000
,:@0.390988:0.404977:0.394191:0.404977:0.394191:0.389264:0.390988:0.389264:0.000000
)= –  :@0.408088:0.405559:0.467629:0.405559:0.467629:0.387905:0.408088:0.387905:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.431576:0.404977:0.457751:0.404977:0.457751:0.388570:0.431576:0.388570:0.000000:0.000000:0.000000
,:@0.458782:0.404977:0.461984:0.404977:0.461984:0.389264:0.458782:0.389264:0.000000
si:@0.485554:0.404977:0.497298:0.404977:0.497298:0.389264:0.485554:0.389264:0.000000:0.000000
 :@0.498310:0.404977:0.502213:0.404977:0.502213:0.388307:0.498310:0.388307:0.000000
x:@0.503243:0.404977:0.510698:0.404977:0.510698:0.388570:0.503243:0.388570:0.000000
 ≥:@0.511729:0.405559:0.531664:0.405559:0.531664:0.387905:0.511729:0.387905:0.000000:0.000000
  ,:@0.532677:0.404977:0.550458:0.404977:0.550458:0.389264:0.532677:0.389264:0.000000:0.000000:0.000000
y:@0.538751:0.404977:0.546224:0.404977:0.546224:0.388570:0.538751:0.388570:0.000000
d x  y:@0.365060:0.425196:0.407029:0.425196:0.407029:0.408790:0.365060:0.408790:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.375350:0.425779:0.381479:0.425779:0.381479:0.408125:0.375350:0.408125:0.000000
,:@0.390978:0.425196:0.394180:0.425196:0.394180:0.409484:0.390978:0.409484:0.000000
)= –  :@0.408078:0.425779:0.467629:0.425774:0.467629:0.408120:0.408078:0.408125:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y x:@0.431566:0.425196:0.457741:0.425196:0.457741:0.408790:0.431566:0.408790:0.000000:0.000000:0.000000
,:@0.458772:0.425196:0.461974:0.425196:0.461974:0.409484:0.458772:0.409484:0.000000
si:@0.485554:0.425191:0.497298:0.425191:0.497298:0.409479:0.485554:0.409479:0.000000:0.000000
 :@0.498310:0.425191:0.502213:0.425191:0.502213:0.408522:0.498310:0.408522:0.000000
x:@0.503243:0.425191:0.510698:0.425191:0.510698:0.408785:0.503243:0.408785:0.000000
 <:@0.511729:0.425774:0.531664:0.425774:0.531664:0.408120:0.511729:0.408120:0.000000:0.000000
  ,:@0.532677:0.425191:0.550458:0.425191:0.550458:0.409479:0.532677:0.409479:0.000000:0.000000:0.000000
y:@0.538751:0.425191:0.546224:0.425191:0.546224:0.408785:0.538751:0.408785:0.000000
A la distancia entre dos números reales la denominaremos métrica usual en   y, como :@0.293658:0.449530:0.930923:0.449530:0.930923:0.433817:0.293658:0.433817:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ℝ:@0.851582:0.450112:0.864872:0.450112:0.864872:0.432458:0.851582:0.432458:0.000000
se dijo, esta nos permite medir la proximidad o lejanía entre dos números reales.:@0.293658:0.466172:0.873478:0.466172:0.873478:0.450459:0.293658:0.450459:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1.  Para :@0.293660:0.489960:0.358423:0.489960:0.358423:0.474247:0.293660:0.474247:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.358320:0.489960:0.365775:0.489960:0.365775:0.473553:0.358320:0.473553:0.000000
=–:@0.366806:0.490542:0.391250:0.490542:0.391250:0.472888:0.366806:0.472888:0.000000:0.000000
3, :@0.391158:0.489960:0.407228:0.489960:0.407228:0.474247:0.391158:0.474247:0.000000:0.000000:0.000000
y:@0.407136:0.489960:0.414609:0.489960:0.414609:0.473553:0.407136:0.473553:0.000000
=:@0.415640:0.490542:0.430973:0.490542:0.430973:0.472888:0.415640:0.472888:0.000000
1, se tiene :@0.430881:0.489960:0.507400:0.489960:0.507400:0.474247:0.430881:0.474247:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d:@0.507234:0.489960:0.516493:0.489960:0.516493:0.473553:0.507234:0.473553:0.000000
(–:@0.517524:0.490542:0.533870:0.490542:0.533870:0.472888:0.517524:0.472888:0.000000:0.000000
3, 1:@0.534882:0.489960:0.563174:0.489960:0.563174:0.474247:0.534882:0.474247:0.000000:0.000000:0.000000:0.000000
)=∙– – ∙ ∙– ∙=:@0.564168:0.490542:0.701118:0.490542:0.701118:0.472888:0.564168:0.472888:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
3  1 = 4:@0.602971:0.489960:0.679655:0.489960:0.679655:0.474247:0.602971:0.474247:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
4.:@0.701026:0.489960:0.713138:0.489960:0.713138:0.474247:0.701026:0.474247:0.000000:0.000000
2.  Si :@0.293660:0.510176:0.340133:0.510176:0.340133:0.494463:0.293660:0.494463:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.340041:0.510176:0.347496:0.510176:0.347496:0.493770:0.340041:0.493770:0.000000
=:@0.348527:0.510758:0.363860:0.510758:0.363860:0.493104:0.348527:0.493104:0.000000
4,6:@0.363768:0.510176:0.384789:0.510176:0.384789:0.494463:0.363768:0.494463:0.000000:0.000000:0.000000
=:@0.384697:0.510758:0.400031:0.510758:0.400031:0.493104:0.384697:0.493104:0.000000
y:@0.399939:0.510176:0.407412:0.510176:0.407412:0.493770:0.399939:0.493770:0.000000
, entonces  4,6; 4,6:@0.408443:0.510176:0.561076:0.510176:0.561076:0.494463:0.408443:0.494463:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d:@0.486858:0.510176:0.496116:0.510176:0.496116:0.493770:0.486858:0.493770:0.000000
(:@0.497147:0.510758:0.503277:0.510758:0.503277:0.493104:0.497147:0.493104:0.000000
)=∙:@0.562051:0.510758:0.589625:0.510758:0.589625:0.493104:0.562051:0.493104:0.000000:0.000000:0.000000
4,6 4,6:@0.590638:0.510176:0.648326:0.510176:0.648326:0.494463:0.590638:0.494463:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
–:@0.614880:0.510758:0.624084:0.510758:0.624084:0.493104:0.614880:0.493104:0.000000
∙=:@0.649338:0.510758:0.669771:0.510758:0.669771:0.493104:0.649338:0.493104:0.000000:0.000000
0.:@0.670783:0.510176:0.683999:0.510176:0.683999:0.494463:0.670783:0.494463:0.000000:0.000000
3. :@0.293660:0.530392:0.309730:0.530392:0.309730:0.514679:0.293660:0.514679:0.000000:0.000000:0.000000
d:@0.323861:0.530392:0.333120:0.530392:0.333120:0.513986:0.323861:0.513986:0.000000
(–:@0.334151:0.530974:0.350497:0.530974:0.350497:0.513320:0.334151:0.513320:0.000000:0.000000
8,  2):@0.351509:0.530392:0.395907:0.530392:0.395907:0.514679:0.351509:0.514679:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
– =∙– –(– )∙=∙– + ∙=∙– ∙=:@0.370800:0.530974:0.628704:0.530974:0.628704:0.513320:0.370800:0.513320:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
8 :@0.428580:0.530392:0.442625:0.530392:0.442625:0.514679:0.428580:0.514679:0.000000:0.000000
2:@0.471211:0.530392:0.480194:0.530392:0.480194:0.514679:0.471211:0.514679:0.000000
8 2:@0.525126:0.530392:0.560468:0.530392:0.560468:0.514679:0.525126:0.514679:0.000000:0.000000:0.000000
6:@0.598258:0.530392:0.607241:0.530392:0.607241:0.514679:0.598258:0.514679:0.000000
6.:@0.629716:0.530392:0.642933:0.530392:0.642933:0.514679:0.629716:0.514679:0.000000:0.000000
3.2 Propiedades de la métrica usual en:@0.293674:0.561332:0.595070:0.561332:0.595070:0.544385:0.293674:0.544385:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.594908:0.561332:0.598811:0.561332:0.598811:0.544662:0.594908:0.544662:0.000000
ℝ:@0.598737:0.561914:0.612027:0.561914:0.612027:0.544260:0.598737:0.544260:0.000000
Teorema. Sean  , ,:@0.293692:0.585116:0.430955:0.585116:0.430955:0.569403:0.293692:0.569403:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x  y  z :@0.403270:0.585116:0.446803:0.585116:0.446803:0.568709:0.403270:0.568709:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∈ ℝ:@0.447853:0.585698:0.482090:0.585698:0.482090:0.568044:0.447853:0.568044:0.000000:0.000000:0.000000
. Se verifican las siguientes propiedades.:@0.483103:0.585116:0.771518:0.585116:0.771518:0.569403:0.483103:0.569403:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a. :@0.293660:0.605333:0.309104:0.605333:0.309104:0.589620:0.293660:0.589620:0.000000:0.000000:0.000000
d x  y:@0.323861:0.605333:0.365830:0.605333:0.365830:0.588927:0.323861:0.588927:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.334151:0.605915:0.340281:0.605915:0.340281:0.588261:0.334151:0.588261:0.000000
,:@0.349779:0.605333:0.352982:0.605333:0.352982:0.589620:0.349779:0.589620:0.000000
)≥:@0.366879:0.605915:0.389354:0.605915:0.389354:0.588261:0.366879:0.588261:0.000000:0.000000
0.:@0.389262:0.605333:0.401374:0.605333:0.401374:0.589620:0.389262:0.589620:0.000000:0.000000
b. :@0.293660:0.627335:0.310558:0.627335:0.310558:0.611623:0.293660:0.611623:0.000000:0.000000:0.000000
d x  y:@0.323861:0.627335:0.365830:0.627335:0.365830:0.610929:0.323861:0.610929:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.334151:0.627918:0.340281:0.627918:0.340281:0.610264:0.334151:0.610264:0.000000
,:@0.349779:0.627335:0.352982:0.627335:0.352982:0.611623:0.349779:0.611623:0.000000
)=:@0.366879:0.627918:0.389354:0.627918:0.389354:0.610264:0.366879:0.610264:0.000000:0.000000
0:@0.390367:0.627335:0.399350:0.627335:0.399350:0.611623:0.390367:0.611623:0.000000
x y:@0.420836:0.627335:0.453141:0.627335:0.453141:0.610929:0.420836:0.610929:0.000000:0.000000:0.000000
=:@0.429322:0.627918:0.444655:0.627918:0.444655:0.610264:0.429322:0.610264:0.000000
.:@0.454171:0.627335:0.457374:0.627335:0.457374:0.611623:0.454171:0.611623:0.000000
c. :@0.293660:0.649338:0.308736:0.649338:0.308736:0.633625:0.293660:0.633625:0.000000:0.000000:0.000000
d x  y:@0.323861:0.649338:0.365830:0.649338:0.365830:0.632932:0.323861:0.632932:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.334151:0.649920:0.340281:0.649920:0.340281:0.632266:0.334151:0.632266:0.000000
,:@0.349779:0.649338:0.352982:0.649338:0.352982:0.633625:0.349779:0.633625:0.000000
)≥ (:@0.366879:0.649920:0.410744:0.649920:0.410744:0.632266:0.366879:0.632266:0.000000:0.000000:0.000000:0.000000
 :@0.389262:0.649338:0.393294:0.649338:0.393294:0.633625:0.389262:0.633625:0.000000
d y  x:@0.394324:0.649338:0.436293:0.649338:0.436293:0.632932:0.394324:0.632932:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
,:@0.420260:0.649338:0.423463:0.649338:0.423463:0.633625:0.420260:0.633625:0.000000
):@0.437342:0.649920:0.443472:0.649920:0.443472:0.632266:0.437342:0.632266:0.000000
. :@0.444484:0.649338:0.451718:0.649338:0.451718:0.633625:0.444484:0.633625:0.000000:0.000000
d. :@0.293660:0.671340:0.310540:0.671340:0.310540:0.655628:0.293660:0.655628:0.000000:0.000000:0.000000
d x  z:@0.323861:0.671340:0.365591:0.671340:0.365591:0.654934:0.323861:0.654934:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.334151:0.671923:0.340281:0.671923:0.340281:0.654269:0.334151:0.654269:0.000000
,:@0.349779:0.671340:0.352982:0.671340:0.352982:0.655628:0.349779:0.655628:0.000000
)≤ (:@0.366640:0.671923:0.410504:0.671923:0.410504:0.654269:0.366640:0.654269:0.000000:0.000000:0.000000:0.000000
 :@0.389023:0.671340:0.393054:0.671340:0.393054:0.655628:0.389023:0.655628:0.000000
d x  y:@0.394085:0.671340:0.436054:0.671340:0.436054:0.654934:0.394085:0.654934:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
,:@0.420002:0.671340:0.423205:0.671340:0.423205:0.655628:0.420002:0.655628:0.000000
):@0.437103:0.671923:0.443232:0.671923:0.443232:0.654269:0.437103:0.654269:0.000000
+:@0.444245:0.671340:0.455216:0.671340:0.455216:0.655628:0.444245:0.655628:0.000000
d y  z:@0.456246:0.671340:0.497994:0.671340:0.497994:0.654934:0.456246:0.654934:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.466536:0.671923:0.472666:0.671923:0.472666:0.654269:0.466536:0.654269:0.000000
,:@0.482182:0.671340:0.485385:0.671340:0.485385:0.655628:0.482182:0.655628:0.000000
):@0.499043:0.671923:0.505173:0.671923:0.505173:0.654269:0.499043:0.654269:0.000000
 (desigualdad triangular). :@0.506185:0.671340:0.691283:0.671340:0.691283:0.655628:0.506185:0.655628:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Demostración. Estas propiedades resultan inmediatamente de las propiedades del valor :@0.293655:0.693349:0.931358:0.693349:0.931358:0.677637:0.293655:0.677637:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
absoluto.:@0.293655:0.709991:0.360146:0.709991:0.360146:0.694278:0.293655:0.694278:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a.  Puesto que :@0.293660:0.733774:0.409698:0.733774:0.409698:0.718061:0.293660:0.718061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∙ – ∙≥:@0.409566:0.734356:0.462302:0.734356:0.462302:0.716702:0.409566:0.716702:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.414664:0.733774:0.440840:0.733774:0.440840:0.717368:0.414664:0.717368:0.000000:0.000000:0.000000
0, se sigue que  :@0.463315:0.733774:0.579476:0.733774:0.579476:0.718061:0.463315:0.718061:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x  y:@0.579299:0.733774:0.621267:0.733774:0.621267:0.717368:0.579299:0.717368:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.589589:0.734356:0.595718:0.734356:0.595718:0.716702:0.589589:0.716702:0.000000
,:@0.605216:0.733774:0.608419:0.733774:0.608419:0.718061:0.605216:0.718061:0.000000
)=∙ – ∙≥:@0.622317:0.734356:0.698541:0.734356:0.698541:0.716702:0.622317:0.716702:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.650903:0.733774:0.677078:0.733774:0.677078:0.717368:0.650903:0.717368:0.000000:0.000000:0.000000
0.:@0.699553:0.733774:0.712770:0.733774:0.712770:0.718061:0.699553:0.718061:0.000000:0.000000
b.  De la definición de distancia entre dos números reales, se tiene:@0.293660:0.755061:0.778835:0.755061:0.778835:0.739349:0.293660:0.739349:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x  y:@0.323659:0.776349:0.365627:0.776349:0.365627:0.759943:0.323659:0.759943:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.333949:0.776932:0.340078:0.776932:0.340078:0.759277:0.333949:0.759277:0.000000
,:@0.349576:0.776349:0.352779:0.776349:0.352779:0.760636:0.349576:0.760636:0.000000
)=∙ – ∙:@0.366677:0.776932:0.426555:0.776932:0.426555:0.759277:0.366677:0.759277:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.395263:0.776349:0.421438:0.776349:0.421438:0.759943:0.395263:0.759943:0.000000:0.000000:0.000000
, luego,:@0.427568:0.776349:0.479561:0.776349:0.479561:0.760636:0.427568:0.760636:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x  y:@0.323751:0.798400:0.365720:0.798400:0.365720:0.781993:0.323751:0.781993:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.334041:0.798982:0.340170:0.798982:0.340170:0.781328:0.334041:0.781328:0.000000
,:@0.349668:0.798400:0.352871:0.798400:0.352871:0.782687:0.349668:0.782687:0.000000
)=:@0.366769:0.798982:0.389244:0.798982:0.389244:0.781328:0.366769:0.781328:0.000000:0.000000
0:@0.390256:0.798400:0.399239:0.798400:0.399239:0.782687:0.390256:0.782687:0.000000
∙ – ∙=:@0.420756:0.798976:0.473492:0.798976:0.473492:0.781322:0.420756:0.781322:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y:@0.425854:0.798393:0.452029:0.798393:0.452029:0.781987:0.425854:0.781987:0.000000:0.000000:0.000000
0:@0.474505:0.798393:0.483487:0.798393:0.483487:0.782681:0.474505:0.782681:0.000000
x y 0:@0.504985:0.798393:0.558587:0.798393:0.558587:0.782681:0.504985:0.782681:0.000000:0.000000:0.000000:0.000000:0.000000
– =:@0.513913:0.798976:0.548592:0.798976:0.548592:0.781322:0.513913:0.781322:0.000000:0.000000:0.000000
⇔:@0.559596:0.798116:0.579649:0.798116:0.579649:0.780355:0.559596:0.780355:0.000000
x y.:@0.580708:0.798393:0.618314:0.798393:0.618314:0.782681:0.580708:0.782681:0.000000:0.000000:0.000000:0.000000
=:@0.589635:0.798976:0.604969:0.798976:0.604969:0.781322:0.589635:0.781322:0.000000
Por lo tanto, :@0.323779:0.819681:0.414726:0.819681:0.414726:0.803968:0.323779:0.803968:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x  y:@0.414582:0.819681:0.456551:0.819681:0.456551:0.803275:0.414582:0.803275:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.424872:0.820263:0.431002:0.820263:0.431002:0.802609:0.424872:0.802609:0.000000
,:@0.440500:0.819681:0.443703:0.819681:0.443703:0.803968:0.440500:0.803968:0.000000
)=:@0.457600:0.820263:0.480075:0.820263:0.480075:0.802609:0.457600:0.802609:0.000000:0.000000
0:@0.481088:0.819681:0.490071:0.819681:0.490071:0.803968:0.481088:0.803968:0.000000
x y.:@0.511552:0.819682:0.549158:0.819682:0.549158:0.803970:0.511552:0.803970:0.000000:0.000000:0.000000:0.000000
=:@0.520479:0.820265:0.535813:0.820265:0.535813:0.802610:0.520479:0.802610:0.000000
c.  Tenemos :@0.293658:0.840970:0.394170:0.840970:0.394170:0.825257:0.293658:0.825257:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x  y:@0.394048:0.840970:0.436017:0.840970:0.436017:0.824564:0.394048:0.824564:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.404338:0.841552:0.410468:0.841552:0.410468:0.823898:0.404338:0.823898:0.000000
,:@0.419966:0.840970:0.423169:0.840970:0.423169:0.825257:0.419966:0.825257:0.000000
)=∙– ( – )∙=∙– ∙∙ – ∙:@0.437066:0.841552:0.621746:0.841552:0.621746:0.823898:0.437066:0.823898:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 1 y x:@0.475869:0.840970:0.525310:0.840970:0.525310:0.825257:0.475869:0.825257:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1:@0.570242:0.840970:0.579225:0.840970:0.579225:0.825257:0.570242:0.825257:0.000000
y x d y  x:@0.590454:0.840970:0.676728:0.840970:0.676728:0.824564:0.590454:0.824564:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
=:@0.622758:0.840970:0.633729:0.840970:0.633729:0.825257:0.622758:0.825257:0.000000
(:@0.645050:0.841552:0.651179:0.841552:0.651179:0.823898:0.645050:0.823898:0.000000
,:@0.660696:0.840970:0.663899:0.840970:0.663899:0.825257:0.660696:0.825257:0.000000
):@0.677778:0.841552:0.683907:0.841552:0.683907:0.823898:0.677778:0.823898:0.000000
. Así,:@0.684920:0.840970:0.716317:0.840970:0.716317:0.825257:0.684920:0.825257:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x  y:@0.322131:0.862258:0.364100:0.862258:0.364100:0.845851:0.322131:0.845851:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.332421:0.862840:0.338550:0.862840:0.338550:0.845186:0.332421:0.845186:0.000000
,:@0.348049:0.862258:0.351251:0.862258:0.351251:0.846545:0.348049:0.846545:0.000000
)= (:@0.365149:0.862840:0.405056:0.862840:0.405056:0.845186:0.365149:0.845186:0.000000:0.000000:0.000000:0.000000
d y  x:@0.388637:0.862258:0.430605:0.862258:0.430605:0.845851:0.388637:0.845851:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
,:@0.414572:0.862258:0.417775:0.862258:0.417775:0.846545:0.414572:0.846545:0.000000
):@0.431654:0.862840:0.437784:0.862840:0.437784:0.845186:0.431654:0.845186:0.000000
.:@0.438796:0.862258:0.441999:0.862258:0.441999:0.846545:0.438796:0.846545:0.000000
d.  Por la desigualdad triangular del valor absoluto, se tiene:@0.293658:0.883545:0.727799:0.883545:0.727799:0.867833:0.293658:0.867833:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x  y:@0.322113:0.904833:0.364081:0.904833:0.364081:0.888427:0.322113:0.888427:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.332402:0.905415:0.338532:0.905415:0.338532:0.887761:0.332402:0.887761:0.000000
,:@0.348030:0.904833:0.351233:0.904833:0.351233:0.889120:0.348030:0.889120:0.000000
)=∙ – ∙:@0.365130:0.905415:0.425009:0.905415:0.425009:0.887761:0.365130:0.887761:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x y d y  z:@0.393717:0.904833:0.477047:0.904833:0.477047:0.888427:0.393717:0.888427:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
, :@0.426022:0.904833:0.434287:0.904833:0.434287:0.889120:0.426022:0.889120:0.000000:0.000000
(:@0.445589:0.905415:0.451718:0.905415:0.451718:0.887761:0.445589:0.887761:0.000000
,:@0.461235:0.904833:0.464438:0.904833:0.464438:0.889120:0.461235:0.889120:0.000000
)=∙ – ∙:@0.478096:0.905415:0.537754:0.905415:0.537754:0.887761:0.478096:0.887761:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
y z:@0.506682:0.904833:0.532636:0.904833:0.532636:0.888427:0.506682:0.888427:0.000000:0.000000:0.000000
. Luego,:@0.538766:0.904833:0.594785:0.904833:0.594785:0.889120:0.538766:0.889120:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x  z:@0.322076:0.926121:0.363805:0.926121:0.363805:0.909715:0.322076:0.909715:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.332366:0.926703:0.338495:0.926703:0.338495:0.909049:0.332366:0.909049:0.000000
,:@0.347993:0.926121:0.351196:0.926121:0.351196:0.910408:0.347993:0.910408:0.000000
)=∙ – ∙ ∙ – + – ≤∙ – ∙+∙ – ∙= (:@0.364854:0.926703:0.658119:0.926703:0.658119:0.909049:0.364854:0.909049:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x z:@0.393441:0.926121:0.419377:0.926121:0.419377:0.909715:0.393441:0.909715:0.000000:0.000000:0.000000
=:@0.425506:0.926121:0.436477:0.926121:0.436477:0.910408:0.425506:0.910408:0.000000
x y y z:@0.442607:0.926121:0.512112:0.926121:0.512112:0.909715:0.442607:0.909715:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
|:@0.513143:0.926121:0.517046:0.926121:0.517046:0.910408:0.513143:0.910408:0.000000
x y:@0.539521:0.926121:0.565696:0.926121:0.565696:0.909715:0.539521:0.909715:0.000000:0.000000:0.000000
y z:@0.593270:0.926121:0.619224:0.926121:0.619224:0.909715:0.593270:0.909715:0.000000:0.000000:0.000000
d x  y:@0.641699:0.926121:0.683668:0.926121:0.683668:0.909715:0.641699:0.909715:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
,:@0.667617:0.926121:0.670820:0.926121:0.670820:0.910408:0.667617:0.910408:0.000000
):@0.684717:0.926703:0.690847:0.926703:0.690847:0.909049:0.684717:0.909049:0.000000
+:@0.691859:0.926121:0.702830:0.926121:0.702830:0.910408:0.691859:0.910408:0.000000
d y  z:@0.703861:0.926121:0.745608:0.926121:0.745608:0.909715:0.703861:0.909715:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.714150:0.926703:0.720280:0.926703:0.720280:0.909049:0.714150:0.909049:0.000000
,:@0.729797:0.926121:0.732999:0.926121:0.732999:0.910408:0.729797:0.910408:0.000000
):@0.746658:0.926703:0.752787:0.926703:0.752787:0.909049:0.746658:0.909049:0.000000
. :@0.753800:0.926121:0.761034:0.926121:0.761034:0.910408:0.753800:0.910408:0.000000:0.000000
Conclusión::@0.293545:0.947408:0.382061:0.947408:0.382061:0.930739:0.293545:0.930739:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.381936:0.947408:0.385967:0.947408:0.385967:0.931696:0.381936:0.931696:0.000000
d x  z:@0.385894:0.947408:0.427623:0.947408:0.427623:0.931002:0.385894:0.931002:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.396184:0.947991:0.402313:0.947991:0.402313:0.930337:0.396184:0.930337:0.000000
,:@0.411811:0.947408:0.415014:0.947408:0.415014:0.931696:0.411811:0.931696:0.000000
)≤ (:@0.428672:0.947991:0.468579:0.947991:0.468579:0.930337:0.428672:0.930337:0.000000:0.000000:0.000000:0.000000
d x  y:@0.452160:0.947408:0.494129:0.947408:0.494129:0.931002:0.452160:0.931002:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
,:@0.478077:0.947408:0.481280:0.947408:0.481280:0.931696:0.478077:0.931696:0.000000
)+ (:@0.495178:0.947991:0.535085:0.947991:0.535085:0.930337:0.495178:0.930337:0.000000:0.000000:0.000000:0.000000
d y  z:@0.518665:0.947408:0.560413:0.947408:0.560413:0.931002:0.518665:0.931002:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
,:@0.544601:0.947408:0.547804:0.947408:0.547804:0.931696:0.544601:0.931696:0.000000
):@0.561462:0.947991:0.567592:0.947991:0.567592:0.930337:0.561462:0.930337:0.000000
.:@0.568604:0.947408:0.571807:0.947408:0.571807:0.931696:0.568604:0.931696:0.000000
¿Cómo se define la distancia entre dos números reales?:@0.293526:0.160305:0.693468:0.160305:0.693468:0.144592:0.293526:0.144592:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Simbología:@0.127120:0.753847:0.213575:0.753847:0.213575:0.737178:0.127120:0.737178:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
matemática:@0.125169:0.770489:0.215517:0.770489:0.215517:0.753820:0.125169:0.753820:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Aprende la siguiente :@0.084236:0.793505:0.224670:0.793505:0.224670:0.779221:0.084236:0.779221:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
simbología matemática: :@0.084236:0.808634:0.246554:0.808634:0.246554:0.794350:0.084236:0.794350:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
A la distancia entre dos :@0.084236:0.823763:0.240457:0.823763:0.240457:0.809479:0.084236:0.809479:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
números reales la :@0.084236:0.838892:0.203235:0.838892:0.203235:0.824608:0.084236:0.824608:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
denominamos métrica :@0.084236:0.854021:0.238666:0.854021:0.238666:0.839737:0.084236:0.839737:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
usual en   y se :@0.084236:0.869150:0.187601:0.869150:0.187601:0.854866:0.084236:0.854866:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
ℝ:@0.143156:0.869680:0.155238:0.869680:0.155238:0.853630:0.143156:0.853630:0.000000
representa por :@0.084236:0.884279:0.184761:0.884279:0.184761:0.869995:0.084236:0.869995:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
d x y:@0.084236:0.899408:0.121519:0.899408:0.121519:0.884493:0.084236:0.884493:0.000000:0.000000:0.000000:0.000000:0.000000
(:@0.093590:0.899937:0.099163:0.899937:0.099163:0.883888:0.093590:0.883888:0.000000
, :@0.107295:0.899408:0.114307:0.899408:0.114307:0.885124:0.107295:0.885124:0.000000:0.000000
)= –:@0.122456:0.899937:0.162868:0.899937:0.162868:0.883888:0.122456:0.883888:0.000000:0.000000:0.000000:0.000000
|:@0.142804:0.899408:0.146352:0.899408:0.146352:0.885124:0.142804:0.885124:0.000000
x y:@0.146787:0.899408:0.170081:0.899408:0.170081:0.884493:0.146787:0.884493:0.000000:0.000000:0.000000
|, :@0.171018:0.899408:0.182012:0.899408:0.182012:0.885124:0.171018:0.885124:0.000000:0.000000:0.000000
x:@0.194010:0.899408:0.200787:0.899408:0.200787:0.884494:0.194010:0.884494:0.000000
, y   .:@0.201731:0.899408:0.250828:0.899408:0.250828:0.885124:0.201731:0.885124:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∈ ℝ:@0.216959:0.899938:0.247498:0.899938:0.247498:0.883889:0.216959:0.883889:0.000000:0.000000:0.000000
EBC 1. Explora y comprende el concepto de límite de una sucesión y de una función.:@0.469843:0.971817:0.915255:0.971817:0.915255:0.960390:0.469843:0.960390:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000