{"id":711,"date":"2021-08-31T06:00:46","date_gmt":"2021-08-31T04:00:46","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=711"},"modified":"2021-08-25T12:45:39","modified_gmt":"2021-08-25T10:45:39","slug":"si-a-es-un-punto-de-acumulacion-de-u-entonces","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/si-a-es-un-punto-de-acumulacion-de-u-entonces\/","title":{"rendered":"Si a es un punto de acumulaci\u00f3n de u, entonces \u2200\u03b5>0, \u2200 N, \u2203k\u2265N, |u(k)\u2212a| < \u03b5"},"content":{"rendered":"

Para extraer una subsucesi\u00f3n se aplica una funci\u00f3n de extracci\u00f3n que conserva el orden; por ejemplo, la subsucesi\u00f3n<\/p>\n

\n   u\u2092, u\u2082, u\u2084, u\u2086, ...\n<\/pre>\n
se ha obtenido con la funci\u00f3n de extracci\u00f3n \u03c6 tal que \u03c6(n) = 2*n.<\/p>\n
En Lean, se puede definir que \u03c6 es una funci\u00f3n de extracci\u00f3n por<\/p>\n
\n   def extraccion (\u03c6 : \u2115 \u2192 \u2115) :=\n     \u2200 n m, n < m \u2192 \u03c6 n < \u03c6 m\n<\/pre>\n
Tambi\u00e9n se puede definir que a es un l\u00edmite de u por<\/p>\n
\n   def limite (u : \u2115 \u2192 \u211d) (a : \u211d) :=\n     \u2200 \u03b5 > 0, \u2203 N, \u2200 k \u2265 N, |u k - a| < \u03b5\n<\/pre>\n
Los puntos de acumulaci\u00f3n de una sucesi\u00f3n son los l\u00edmites de sus subsucesiones. En Lean se puede definir por<\/p>\n
\n   def punto_acumulacion (u : \u2115 \u2192 \u211d) (a : \u211d) :=\n     \u2203 \u03c6, extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a\n<\/pre>\n
Demostrar que si a es un punto de acumulaci\u00f3n de u, entonces<\/p>\n
\n   \u2200 \u03b5 > 0, \u2200 N, \u2203 k \u2265 N, |u k - a| < \u03b5\n<\/pre>\n
Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n
\nimport data.real.basic\nopen nat\n\nvariable  {u : \u2115 \u2192 \u211d}\nvariables {a : \u211d}\nvariable  {\u03c6 : \u2115 \u2192 \u2115}\n\ndef extraccion (\u03c6 : \u2115 \u2192 \u2115):=\n  \u2200 n m, n < m \u2192 \u03c6 n < \u03c6 m\n\nnotation `|`x`|` := abs x\n\ndef limite (u : \u2115 \u2192 \u211d) (a : \u211d) :=\n  \u2200 \u03b5 > 0, \u2203 N, \u2200 k \u2265 N, |u k - a| < \u03b5\n\ndef punto_acumulacion (u : \u2115 \u2192 \u211d) (a : \u211d) :=\n  \u2203 \u03c6, extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a\n\nexample\n  (h : punto_acumulacion u a)\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 k \u2265 N, |u k - a| < \u03b5 :=\nsorry\n<\/pre>\n
[expand title=\"Soluciones con Lean\"]<\/p>\n
\r\nimport data.real.basic\r\nopen nat\r\n\r\nvariable  {u : \u2115 \u2192 \u211d}\r\nvariables {a : \u211d}\r\nvariable  {\u03c6 : \u2115 \u2192 \u2115}\r\n\r\ndef extraccion (\u03c6 : \u2115 \u2192 \u2115):=\r\n  \u2200 n m, n < m \u2192 \u03c6 n < \u03c6 m\r\n\r\nnotation `|`x`|` := abs x\r\n\r\ndef limite (u : \u2115 \u2192 \u211d) (a : \u211d) :=\r\n  \u2200 \u03b5 > 0, \u2203 N, \u2200 k \u2265 N, |u k - a| < \u03b5\r\n\r\ndef punto_acumulacion (u : \u2115 \u2192 \u211d) (a : \u211d) :=\r\n  \u2203 \u03c6, extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a\r\n\r\n-- En la demostraci\u00f3n se usar\u00e1n los siguientes lemas.\r\n\r\nlemma aux1\r\n  (h : extraccion \u03c6)\r\n  : \u2200 n, n \u2264 \u03c6 n :=\r\nbegin\r\n  intro n,\r\n  induction n with m HI,\r\n  { exact nat.zero_le (\u03c6 0), },\r\n  { apply nat.succ_le_of_lt,\r\n    calc m \u2264 \u03c6 m        : HI\r\n       ... < \u03c6 (succ m) : h m (m+1) (lt_add_one m), },\r\nend\r\n\r\nlemma aux2\r\n  (h : extraccion \u03c6)\r\n  : \u2200 N N', \u2203 n \u2265 N', \u03c6 n \u2265 N :=\r\n\u03bb N N', \u27e8max N N', \u27e8le_max_right N N',\r\n                    le_trans (le_max_left N N')\r\n                             (aux1 h (max N N'))\u27e9\u27e9\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 k \u2265 N, |u k - a| < \u03b5 :=\r\nbegin\r\n  intros \u03b5 h\u03b5 N,\r\n  unfold punto_acumulacion at h,\r\n  rcases h with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  unfold limite at h\u03c62,\r\n  cases h\u03c62 \u03b5 h\u03b5 with N' hN',\r\n  rcases aux2 h\u03c61 N N' with \u27e8m, hm, hm'\u27e9,\r\n  clear h\u03c61 h\u03c62,\r\n  use \u03c6 m,\r\n  split,\r\n  { exact hm', },\r\n  { exact hN' m hm, },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nbegin\r\n  intros \u03b5 h\u03b5 N,\r\n  rcases h with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  cases h\u03c62 \u03b5 h\u03b5 with N' hN',\r\n  rcases aux2 h\u03c61 N N' with \u27e8m, hm, hm'\u27e9,\r\n  use \u03c6 m,\r\n  exact \u27e8hm', hN' m hm\u27e9,\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nbegin\r\n  intros \u03b5 h\u03b5 N,\r\n  rcases h with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  cases h\u03c62 \u03b5 h\u03b5 with N' hN',\r\n  rcases aux2 h\u03c61 N N' with \u27e8m, hm, hm'\u27e9,\r\n  exact \u27e8\u03c6 m, hm', hN' _ hm\u27e9,\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nbegin\r\n  intros \u03b5 h\u03b5 N,\r\n  rcases h with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  cases h\u03c62 \u03b5 h\u03b5 with N' hN',\r\n  rcases aux2 h\u03c61 N N' with \u27e8m, hm, hm'\u27e9,\r\n  use \u03c6 m ; finish,\r\nend\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  ( assume \u03c6,\r\n    assume h\u03c6 : extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a,\r\n    exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n      ( assume N',\r\n        assume hN' : \u2200 (n : \u2115), n \u2265 N' \u2192 |(u \u2218 \u03c6) n - a| < \u03b5,\r\n        have h1 : \u2203 n \u2265 N', \u03c6 n \u2265 N,\r\n          from aux2 h\u03c6.1 N N',\r\n        exists.elim h1\r\n          ( assume m,\r\n            assume hm : \u2203 (H : m \u2265 N'), \u03c6 m \u2265 N,\r\n            exists.elim hm\r\n              ( assume hm1 : m \u2265 N',\r\n                assume hm2 : \u03c6 m \u2265 N,\r\n                have h2 : |u (\u03c6 m) - a| < \u03b5,\r\n                  from hN' m hm1,\r\n                show \u2203 n \u2265 N, |u n - a| < \u03b5,\r\n                  from exists.intro (\u03c6 m) (exists.intro hm2 h2)))))\r\n\r\n-- 6\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  ( assume \u03c6,\r\n    assume h\u03c6 : extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a,\r\n    exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n      ( assume N',\r\n        assume hN' : \u2200 (n : \u2115), n \u2265 N' \u2192 |(u \u2218 \u03c6) n - a| < \u03b5,\r\n        have h1 : \u2203 n \u2265 N', \u03c6 n \u2265 N,\r\n          from aux2 h\u03c6.1 N N',\r\n        exists.elim h1\r\n          ( assume m,\r\n            assume hm : \u2203 (H : m \u2265 N'), \u03c6 m \u2265 N,\r\n            exists.elim hm\r\n              ( assume hm1 : m \u2265 N',\r\n                assume hm2 : \u03c6 m \u2265 N,\r\n                have h2 : |u (\u03c6 m) - a| < \u03b5,\r\n                  from hN' m hm1,\r\n                show \u2203 n \u2265 N, |u n - a| < \u03b5,\r\n                  from \u27e8\u03c6 m, hm2, h2\u27e9))))\r\n\r\n-- 7\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  ( assume \u03c6,\r\n    assume h\u03c6 : extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a,\r\n    exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n      ( assume N',\r\n        assume hN' : \u2200 (n : \u2115), n \u2265 N' \u2192 |(u \u2218 \u03c6) n - a| < \u03b5,\r\n        have h1 : \u2203 n \u2265 N', \u03c6 n \u2265 N,\r\n          from aux2 h\u03c6.1 N N',\r\n        exists.elim h1\r\n          ( assume m,\r\n            assume hm : \u2203 (H : m \u2265 N'), \u03c6 m \u2265 N,\r\n            exists.elim hm\r\n              ( assume hm1 : m \u2265 N',\r\n                assume hm2 : \u03c6 m \u2265 N,\r\n                have h2 : |u (\u03c6 m) - a| < \u03b5,\r\n                  from hN' m hm1,\r\n                \u27e8\u03c6 m, hm2, h2\u27e9))))\r\n\r\n-- 8\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  ( assume \u03c6,\r\n    assume h\u03c6 : extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a,\r\n    exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n      ( assume N',\r\n        assume hN' : \u2200 (n : \u2115), n \u2265 N' \u2192 |(u \u2218 \u03c6) n - a| < \u03b5,\r\n        have h1 : \u2203 n \u2265 N', \u03c6 n \u2265 N,\r\n          from aux2 h\u03c6.1 N N',\r\n        exists.elim h1\r\n          ( assume m,\r\n            assume hm : \u2203 (H : m \u2265 N'), \u03c6 m \u2265 N,\r\n            exists.elim hm\r\n              ( assume hm1 : m \u2265 N',\r\n                assume hm2 : \u03c6 m \u2265 N,\r\n                \u27e8\u03c6 m, hm2, hN' m hm1\u27e9))))\r\n\r\n-- 9\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  ( assume \u03c6,\r\n    assume h\u03c6 : extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a,\r\n    exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n      ( assume N',\r\n        assume hN' : \u2200 (n : \u2115), n \u2265 N' \u2192 |(u \u2218 \u03c6) n - a| < \u03b5,\r\n        have h1 : \u2203 n \u2265 N', \u03c6 n \u2265 N,\r\n          from aux2 h\u03c6.1 N N',\r\n        exists.elim h1\r\n          ( assume m,\r\n            assume hm : \u2203 (H : m \u2265 N'), \u03c6 m \u2265 N,\r\n            exists.elim hm\r\n              (\u03bb hm1 hm2, \u27e8\u03c6 m, hm2, hN' m hm1\u27e9))))\r\n\r\n-- 10\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  ( assume \u03c6,\r\n    assume h\u03c6 : extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a,\r\n    exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n      ( assume N',\r\n        assume hN' : \u2200 (n : \u2115), n \u2265 N' \u2192 |(u \u2218 \u03c6) n - a| < \u03b5,\r\n        have h1 : \u2203 n \u2265 N', \u03c6 n \u2265 N,\r\n          from aux2 h\u03c6.1 N N',\r\n        exists.elim h1\r\n          (\u03bb m hm, exists.elim hm (\u03bb hm1 hm2, \u27e8\u03c6 m, hm2, hN' m hm1\u27e9))))\r\n\r\n-- 11\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  ( assume \u03c6,\r\n    assume h\u03c6 : extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a,\r\n    exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n      ( assume N',\r\n        assume hN' : \u2200 (n : \u2115), n \u2265 N' \u2192 |(u \u2218 \u03c6) n - a| < \u03b5,\r\n        exists.elim (aux2 h\u03c6.1 N N')\r\n          (\u03bb m hm, exists.elim hm (\u03bb hm1 hm2, \u27e8\u03c6 m, hm2, hN' m hm1\u27e9))))\r\n\r\n-- 12\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  ( assume \u03c6,\r\n    assume h\u03c6 : extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a,\r\n    exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n      (\u03bb N' hN', exists.elim (aux2 h\u03c6.1 N N')\r\n        (\u03bb m hm, exists.elim hm\r\n          (\u03bb hm1 hm2, \u27e8\u03c6 m, hm2, hN' m hm1\u27e9))))\r\n\r\n-- 13\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\nassume \u03b5,\r\nassume h\u03b5 : \u03b5 > 0,\r\nassume N,\r\nexists.elim h\r\n  (\u03bb \u03c6 h\u03c6, exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n    (\u03bb N' hN', exists.elim (aux2 h\u03c6.1 N N')\r\n      (\u03bb m hm, exists.elim hm\r\n        (\u03bb hm1 hm2, \u27e8\u03c6 m, hm2, hN' m hm1\u27e9))))\r\n\r\n-- 14\u00aa demostraci\u00f3n\r\nexample\r\n  (h : punto_acumulacion u a)\r\n  : \u2200 \u03b5 > 0, \u2200 N, \u2203 n \u2265 N, |u n - a| < \u03b5 :=\r\n\u03bb \u03b5 h\u03b5 N, exists.elim h\r\n  (\u03bb \u03c6 h\u03c6, exists.elim (h\u03c6.2 \u03b5 h\u03b5)\r\n    (\u03bb N' hN', exists.elim (aux2 h\u03c6.1 N N')\r\n      (\u03bb m hm, exists.elim hm\r\n        (\u03bb hm1 hm2, \u27e8\u03c6 m, hm2, hN' m hm1\u27e9))))\r\n<\/pre>\n
Se puede interactuar con la prueba anterior en esta sesi\u00f3n con Lean<\/a>.<\/p>\n
En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con <pre lang="lean"> y otra con <\/pre>
\n[\/expand]<\/p>\n
[expand title=\"Soluciones con Isabelle\/HOL\"]<\/p>\n
\r\ntheory \"Si_a_es_un_punto_de_acumulacion_de_u,_entonces_a_tiene_puntos_cercanos\"\r\nimports Main HOL.Real\r\nbegin\r\n\r\ndefinition extraccion :: \"(nat \u21d2 nat) \u21d2 bool\" where\r\n  \"extraccion \u03c6 \u27f7 (\u2200 n m. n < m \u27f6 \u03c6 n < \u03c6 m)\"\r\n\r\ndefinition limite :: \"(nat \u21d2 real) \u21d2 real \u21d2 bool\"\r\n  where \"limite u a \u27f7 (\u2200\u03b5>0. \u2203N. \u2200k\u2265N. \u00a6u k - a\u00a6 < \u03b5)\"\r\n\r\ndefinition punto_acumulacion :: \"(nat \u21d2 real) \u21d2 real \u21d2 bool\"\r\n  where \"punto_acumulacion u a \u27f7 (\u2203\u03c6. extraccion \u03c6 \u2227 limite (u \u2218 \u03c6) a)\"\r\n\r\n(* En la demostraci\u00f3n se usar\u00e1n los siguientes lemas *)\r\nlemma aux1 :\r\n  assumes \"extraccion \u03c6\"\r\n  shows   \"n \u2264 \u03c6 n\"\r\nproof (induct n)\r\n  show \"0 \u2264 \u03c6 0\" by simp\r\nnext\r\n  fix n assume HI : \"n \u2264 \u03c6 n\"\r\n  then show \"Suc n \u2264 \u03c6 (Suc n)\"\r\n    using assms extraccion_def\r\n    by (metis Suc_leI lessI order_le_less_subst1)\r\nqed\r\n\r\nlemma aux2 :\r\n  assumes \"extraccion \u03c6\"\r\n  shows   \"\u2200 N N'. \u2203 k \u2265 N'. \u03c6 k \u2265 N\"\r\nproof (intro allI)\r\n  fix N N' :: nat\r\n  have \"max N N' \u2265 N' \u2227 \u03c6 (max N N') \u2265 N\"\r\n    by (meson assms aux1 max.bounded_iff max.cobounded2)\r\n  then show \"\u2203k \u2265 N'. \u03c6 k \u2265 N\"\r\n    by blast\r\nqed\r\n\r\n(* 1\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"punto_acumulacion u a\"\r\n  shows   \"\u2200\u03b5>0. \u2200 N. \u2203k\u2265N. \u00a6u k - a\u00a6 < \u03b5\"\r\nproof (intro allI impI)\r\n  fix \u03b5 :: real and N :: nat\r\n  assume \"\u03b5 > 0\"\r\n  obtain \u03c6 where h\u03c61 : \"extraccion \u03c6\"\r\n             and h\u03c62 : \"limite (u \u2218 \u03c6) a\"\r\n    using assms punto_acumulacion_def by blast\r\n  obtain N' where hN' : \"\u2200k\u2265N'. \u00a6(u \u2218 \u03c6) k - a\u00a6 < \u03b5\"\r\n    using h\u03c62 limite_def \u2039\u03b5 > 0\u203a by auto\r\n  obtain m where hm1 : \"m \u2265 N'\" and hm2 : \"\u03c6 m \u2265 N\"\r\n    using aux2 h\u03c61 by blast\r\n  have \"\u03c6 m \u2265 N \u2227 \u00a6u (\u03c6 m) - a\u00a6 < \u03b5\"\r\n    using hN' hm1 hm2 by force\r\n  then show \"\u2203k\u2265N. \u00a6u k - a\u00a6 < \u03b5\"\r\n    by auto\r\nqed\r\n\r\n(* 2\u00aa demostraci\u00f3n *)\r\nlemma\r\n  assumes \"punto_acumulacion u a\"\r\n  shows   \"\u2200\u03b5>0. \u2200 N. \u2203k\u2265N. \u00a6u k - a\u00a6 < \u03b5\"\r\nproof (intro allI impI)\r\n  fix \u03b5 :: real and N :: nat\r\n  assume \"\u03b5 > 0\"\r\n  obtain \u03c6 where h\u03c61 : \"extraccion \u03c6\"\r\n             and h\u03c62 : \"limite (u \u2218 \u03c6) a\"\r\n    using assms punto_acumulacion_def by blast\r\n  obtain N' where hN' : \"\u2200k\u2265N'. \u00a6(u \u2218 \u03c6) k - a\u00a6 < \u03b5\"\r\n    using h\u03c62 limite_def \u2039\u03b5 > 0\u203a by auto\r\n  obtain m where \"m \u2265 N' \u2227 \u03c6 m \u2265 N\"\r\n    using aux2 h\u03c61 by blast\r\n  then show \"\u2203k\u2265N. \u00a6u k - a\u00a6 < \u03b5\"\r\n    using hN' by auto\r\nqed\r\n\r\nend\r\n<\/pre>\n
En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con <pre lang="isar"> y otra con <\/pre>
\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"
Para extraer una subsucesi\u00f3n se aplica una funci\u00f3n de extracci\u00f3n que conserva el orden; por ejemplo, la subsucesi\u00f3n u\u2092, u\u2082, u\u2084, u\u2086, … se ha obtenido con la funci\u00f3n de extracci\u00f3n \u03c6 tal que \u03c6(n) = 2*n. En Lean, se puede definir que \u03c6 es una funci\u00f3n de extracci\u00f3n por def extraccion (\u03c6 : \u2115 \u2192 \u2115) := \u2200 n m, n < m \u2192 \u03c6 n < \u03c6 m Tambi\u00e9n se puede definir que a es un l\u00edmite de u por def limite (u : \u2115 \u2192 \u211d) (a : \u211d) := \u2200 \u03b5 > 0, \u2203 N, \u2200 k \u2265 N, |u k – a| < \u03b5 Los puntos de acumulaci\u00f3n de una sucesi\u00f3n son los l\u00edmites de sus subsucesiones. En Lean se...\n<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[14],"tags":[90,77,92,192,91,177,115,179,186,191,76,60,190,100,180,80,81,49,188,189,125,169,43,63,183,182,173,181,171,170,45,48,187,46],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/711"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/comments?post=711"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/711\/revisions"}],"predecessor-version":[{"id":716,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/711\/revisions\/716"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=711"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=711"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=711"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}