equal
deleted
inserted
replaced
2338 apply simp |
2338 apply simp |
2339 done |
2339 done |
2340 |
2340 |
2341 lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))" |
2341 lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))" |
2342 by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all |
2342 by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all |
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2343 |
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2344 lemma uniformly_continuous_imp_Cauchy_continuous: |
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2345 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" |
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2346 shows "\<lbrakk>uniformly_continuous_on S f; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f o \<sigma>)" |
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2347 by (simp add: uniformly_continuous_on_def Cauchy_def) meson |
2343 |
2348 |
2344 lemma (in bounded_linear) isUCont: "isUCont f" |
2349 lemma (in bounded_linear) isUCont: "isUCont f" |
2345 unfolding isUCont_def dist_norm |
2350 unfolding isUCont_def dist_norm |
2346 proof (intro allI impI) |
2351 proof (intro allI impI) |
2347 fix r :: real |
2352 fix r :: real |