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Limite_de_la_suma_de_sucesiones_convergentes.thy
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(* Limite_de_la_suma_de_sucesiones_convergentes.thy
-- Límite de la suma de sucesiones convergentes
-- José A. Alonso Jiménez
-- Sevilla, 14 de julio de 2021
-- ------------------------------------------------------------------ *)
(* ---------------------------------------------------------------------
-- En Isabelle/HOL, una sucesión u₀, u₁, u₂, ... se puede representar
-- mediante una función (u : \<nat> \<rightarrow> \<real>) de forma que u(n) es uₙ.
--
-- Se define que a es el límite de la sucesión u, por
-- definition limite :: "(nat \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool"
-- where "limite u c \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>k::nat. \<forall>n\<ge>k. \<bar>u n - c\<bar> < \<epsilon>)"
--
-- Demostrar que el límite de la suma de dos sucesiones convergentes es
-- la suma de los límites de dichas sucesiones.
-- ------------------------------------------------------------------ *)
theory Limite_de_la_suma_de_sucesiones_convergentes
imports Main HOL.Real
begin
definition limite :: "(nat \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool"
where "limite u c \<longleftrightarrow> (\<forall>\<epsilon>>0. \<exists>k::nat. \<forall>n\<ge>k. \<bar>u n - c\<bar> < \<epsilon>)"
(* 1\<ordfeminine> demostración *)
lemma
assumes "limite u a"
"limite v b"
shows "limite (\<lambda> n. u n + v n) (a + b)"
proof (unfold limite_def)
show "\<forall>\<epsilon>>0. \<exists>k. \<forall>n\<ge>k. \<bar>(u n + v n) - (a + b)\<bar> < \<epsilon>"
proof (intro allI impI)
fix \<epsilon> :: real
assume "0 < \<epsilon>"
then have "0 < \<epsilon>/2"
by simp
then have "\<exists>k. \<forall>n\<ge>k. \<bar>u n - a\<bar> < \<epsilon>/2"
using assms(1) limite_def by blast
then obtain Nu where hNu : "\<forall>n\<ge>Nu. \<bar>u n - a\<bar> < \<epsilon>/2"
by (rule exE)
then have "\<exists>k. \<forall>n\<ge>k. \<bar>v n - b\<bar> < \<epsilon>/2"
using \<open>0 < \<epsilon>/2\<close> assms(2) limite_def by blast
then obtain Nv where hNv : "\<forall>n\<ge>Nv. \<bar>v n - b\<bar> < \<epsilon>/2"
by (rule exE)
have "\<forall>n\<ge>max Nu Nv. \<bar>(u n + v n) - (a + b)\<bar> < \<epsilon>"
proof (intro allI impI)
fix n :: nat
assume "n \<ge> max Nu Nv"
have "\<bar>(u n + v n) - (a + b)\<bar> = \<bar>(u n - a) + (v n - b)\<bar>"
by simp
also have "\<dots> \<le> \<bar>u n - a\<bar> + \<bar>v n - b\<bar>"
by simp
also have "\<dots> < \<epsilon>/2 + \<epsilon>/2"
using hNu hNv \<open>max Nu Nv \<le> n\<close> by fastforce
finally show "\<bar>(u n + v n) - (a + b)\<bar> < \<epsilon>"
by simp
qed
then show "\<exists>k. \<forall>n\<ge>k. \<bar>u n + v n - (a + b)\<bar> < \<epsilon> "
by (rule exI)
qed
qed
(* 2\<ordfeminine> demostración *)
lemma
assumes "limite u a"
"limite v b"
shows "limite (\<lambda> n. u n + v n) (a + b)"
proof (unfold limite_def)
show "\<forall>\<epsilon>>0. \<exists>k. \<forall>n\<ge>k. \<bar>(u n + v n) - (a + b)\<bar> < \<epsilon>"
proof (intro allI impI)
fix \<epsilon> :: real
assume "0 < \<epsilon>"
then have "0 < \<epsilon>/2" by simp
obtain Nu where hNu : "\<forall>n\<ge>Nu. \<bar>u n - a\<bar> < \<epsilon>/2"
using \<open>0 < \<epsilon>/2\<close> assms(1) limite_def by blast
obtain Nv where hNv : "\<forall>n\<ge>Nv. \<bar>v n - b\<bar> < \<epsilon>/2"
using \<open>0 < \<epsilon>/2\<close> assms(2) limite_def by blast
have "\<forall>n\<ge>max Nu Nv. \<bar>(u n + v n) - (a + b)\<bar> < \<epsilon>"
using hNu hNv
by (smt (verit, ccfv_threshold) field_sum_of_halves max.boundedE)
then show "\<exists>k. \<forall>n\<ge>k. \<bar>u n + v n - (a + b)\<bar> < \<epsilon> "
by blast
qed
qed
end