{"id":714,"date":"2021-09-01T06:00:52","date_gmt":"2021-09-01T04:00:52","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/?p=714"},"modified":"2021-08-25T12:42:53","modified_gmt":"2021-08-25T10:42:53","slug":"las-subsucesiones-tienen-el-mismo-limite-que-la-sucesion","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/las-subsucesiones-tienen-el-mismo-limite-que-la-sucesion\/","title":{"rendered":"Las subsucesiones tienen el mismo l\u00edmite que la sucesi\u00f3n"},"content":{"rendered":"<p>Para extraer una subsucesi\u00f3n se aplica una funci\u00f3n de extracci\u00f3n que conserva el orden; por ejemplo, la subsucesi\u00f3n<\/p>\n<pre lang=\"text\">\n   u\u2092, u\u2082, u\u2084, u\u2086, ...\n<\/pre>\n<p>se ha obtenido con la funci\u00f3n de extracci\u00f3n \u03c6 tal que \u03c6(n) = 2*n.<\/p>\n<p>En Lean, se puede definir que \u03c6 es una funci\u00f3n de extracci\u00f3n por<\/p>\n<pre lang=\"text\">\n   def extraccion (\u03c6 : \u2115 \u2192 \u2115) :=\n     \u2200 n m, n < m \u2192 \u03c6 n < \u03c6 m\n<\/pre>\n<p>que v es una subsucesi\u00f3n de u por<\/p>\n<pre lang=\"text\">\n   def subsucesion (v u : \u2115 \u2192 \u211d) :=\n     \u2203 \u03c6, extraccion \u03c6 \u2227 v = u \u2218 \u03c6\n<\/pre>\n<p>y que a es un l\u00edmite de u por<\/p>\n<pre lang=\"text\">\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<p>Demostrar que las subsucesiones de una sucesi\u00f3n convergente tienen el mismo l\u00edmite que la sucesi\u00f3n.<\/p>\n<p>Para ello, completar la siguiente teor\u00eda de Lean:<\/p>\n<pre lang=\"lean\">\nimport data.real.basic\nopen nat\n\nvariables {u v : \u2115 \u2192 \u211d}\nvariable  {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\ndef subsucesion (v u : \u2115 \u2192 \u211d) :=\n  \u2203 \u03c6, extraccion \u03c6 \u2227 v = u \u2218 \u03c6\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\nexample\n  (hv : subsucesion v u)\n  (ha : limite u a)\n  : limite v a :=\nsorry\n<\/pre>\n<p>[expand title=\"Soluciones con Lean\"]<\/p>\n<pre lang=\"lean\">\r\nimport data.real.basic\r\nopen nat\r\n\r\nvariables {u v : \u2115 \u2192 \u211d}\r\nvariable  {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\ndef subsucesion (v u : \u2115 \u2192 \u211d) :=\r\n  \u2203 \u03c6, extraccion \u03c6 \u2227 v = u \u2218 \u03c6\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\n-- En la demostraci\u00f3n se usar\u00e1 el siguiente lema.\r\nlemma aux\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\n-- 1\u00aa demostraci\u00f3n\r\nexample\r\n  (hv : subsucesion v u)\r\n  (ha : limite u a)\r\n  : limite v a :=\r\nbegin\r\n  unfold limite,\r\n  intros \u03b5 h\u03b5,\r\n  unfold limite at ha,\r\n  cases ha \u03b5 h\u03b5 with N hN,\r\n  use N,\r\n  intros n hn,\r\n  unfold subsucesion at hv,\r\n  rcases hv with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  rw h\u03c62,\r\n  apply hN,\r\n  apply le_trans hn,\r\n  exact aux h\u03c61 n,\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n  (hv : subsucesion v u)\r\n  (ha : limite u a)\r\n  : limite v a :=\r\nbegin\r\n  intros \u03b5 h\u03b5,\r\n  cases ha \u03b5 h\u03b5 with N hN,\r\n  use N,\r\n  intros n hn,\r\n  rcases hv with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  rw h\u03c62,\r\n  apply hN,\r\n  exact le_trans hn (aux h\u03c61 n),\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\nexample\r\n  (hv : subsucesion v u)\r\n  (ha : limite u a)\r\n  : limite v a :=\r\nbegin\r\n  intros \u03b5 h\u03b5,\r\n  cases ha \u03b5 h\u03b5 with N hN,\r\n  use N,\r\n  intros n hn,\r\n  rcases hv with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  rw h\u03c62,\r\n  exact hN (\u03c6 n) (le_trans hn (aux h\u03c61 n)),\r\nend\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\nexample\r\n  (hv : subsucesion v u)\r\n  (ha : limite u a)\r\n  : limite v a :=\r\nbegin\r\n  intros \u03b5 h\u03b5,\r\n  cases ha \u03b5 h\u03b5 with N hN,\r\n  rcases hv with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  rw h\u03c62,\r\n  use N,\r\n  exact \u03bb n hn, hN (\u03c6 n) (le_trans hn (aux h\u03c61 n)),\r\nend\r\n\r\n-- 5\u00aa demostraci\u00f3n\r\nexample\r\n  (hv : subsucesion v u)\r\n  (ha : limite u a)\r\n  : limite v a :=\r\nbegin\r\n  intros \u03b5 h\u03b5,\r\n  cases ha \u03b5 h\u03b5 with N hN,\r\n  rcases hv with \u27e8\u03c6, h\u03c61, h\u03c62\u27e9,\r\n  rw h\u03c62,\r\n  exact \u27e8N, \u03bb n hn, hN (\u03c6 n) (le_trans hn (aux h\u03c61 n))\u27e9,\r\nend\r\n<\/pre>\n<p>Se puede interactuar con la prueba anterior en <a href=\"https:\/\/leanprover-community.github.io\/lean-web-editor\/#url=https:\/\/raw.githubusercontent.com\/jaalonso\/Calculemus\/main\/src\/Las_subsucesiones_tienen_el_mismo_limite_que_la_sucesion.lean\" rel=\"noopener noreferrer\" target=\"_blank\">esta sesi\u00f3n con Lean<\/a>.<\/p>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;lean&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n<p>[expand title=\"Soluciones con Isabelle\/HOL\"]<\/p>\n<pre lang=\"isar\">\r\ntheory Las_subsucesiones_tienen_el_mismo_limite_que_la_sucesion\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 subsucesion :: \"(nat \u21d2 real) \u21d2 (nat \u21d2 real) \u21d2 bool\"\r\n  where \"subsucesion v u \u27f7 (\u2203 \u03c6. extraccion \u03c6 \u2227 v = u \u2218 \u03c6)\"\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\n(* En la demostraci\u00f3n se usar\u00e1 el siguiente lema *)\r\nlemma aux :\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\n(* Demostraci\u00f3n *)\r\nlemma\r\n  assumes \"subsucesion v u\"\r\n          \"limite u a\"\r\n  shows   \"limite v a\"\r\nproof (unfold limite_def; intro allI impI)\r\n  fix \u03b5 :: real\r\n  assume \"\u03b5 > 0\"\r\n  then obtain N where hN : \"\u2200k\u2265N. \u00a6u k - a\u00a6 < \u03b5\"\r\n    using assms(2) limite_def by auto\r\n  obtain \u03c6 where h\u03c61 : \"extraccion \u03c6\" and h\u03c62 : \"v = u \u2218 \u03c6\"\r\n    using assms(1) subsucesion_def by auto\r\n  have \"\u2200k\u2265N. \u00a6v k - a\u00a6 < \u03b5\"\r\n  proof (intro allI impI)\r\n    fix k\r\n    assume \"N \u2264 k\"\r\n    also have \"... \u2264 \u03c6 k\"\r\n      by (simp add: aux h\u03c61)\r\n    finally have \"N \u2264 \u03c6 k\" .\r\n    have \"\u00a6v k - a\u00a6 = \u00a6u (\u03c6 k) - a\u00a6\"\r\n      using h\u03c62 by simp\r\n    also have \"\u2026 < \u03b5\"\r\n      using hN \u2039N \u2264 \u03c6 k\u203a by simp\r\n    finally show \"\u00a6v k - a\u00a6 < \u03b5\" .\r\n  qed\r\n  then show \"\u2203N. \u2200k\u2265N. \u00a6v k - a\u00a6 < \u03b5\"\r\n    by auto\r\nqed\r\n\r\nend\r\n<\/pre>\n<p>En los comentarios se pueden escribir otras soluciones, escribiendo el c\u00f3digo entre una l\u00ednea con &#60;pre lang=&quot;isar&quot;&#62; y otra con &#60;\/pre&#62;<br \/>\n[\/expand]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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, &#8230; 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 que v es una subsucesi\u00f3n de u por def subsucesion (v u : \u2115 \u2192 \u211d) := \u2203 \u03c6, extraccion \u03c6 \u2227 v = u \u2218 \u03c6 y que a es un l\u00edmite de u por def limite (u : \u2115 \u2192 \u211d) (a : \u211d) := \u2200 \u03b5 > 0, \u2203 N, \u2200 k&#8230;<\/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":[90,77,91,177,115,179,60,190,100,180,114,109,80,49,169,43,63,184,173,171,170,45,111,187,46],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/714"}],"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=714"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/714\/revisions"}],"predecessor-version":[{"id":715,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/posts\/714\/revisions\/715"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/media?parent=714"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/categories?post=714"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/calculemus\/wp-json\/wp\/v2\/tags?post=714"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}