equal
deleted
inserted
replaced
2947 unfolding uniformly_continuous_on_def by (meson Cauchy_def) |
2947 unfolding uniformly_continuous_on_def by (meson Cauchy_def) |
2948 |
2948 |
2949 lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))" |
2949 lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))" |
2950 by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all |
2950 by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all |
2951 |
2951 |
2952 lemma uniformly_continuous_imp_Cauchy_continuous: |
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2953 fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" |
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2954 shows "\<lbrakk>uniformly_continuous_on S f; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f \<circ> \<sigma>)" |
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2955 by (simp add: uniformly_continuous_on_def Cauchy_def) meson |
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2956 |
|
2957 lemma (in bounded_linear) isUCont: "isUCont f" |
2952 lemma (in bounded_linear) isUCont: "isUCont f" |
2958 unfolding isUCont_def dist_norm |
2953 unfolding isUCont_def dist_norm |
2959 proof (intro allI impI) |
2954 proof (intro allI impI) |
2960 fix r :: real |
2955 fix r :: real |
2961 assume r: "0 < r" |
2956 assume r: "0 < r" |