src/HOL/Transcendental.thy
 changeset 44726 8478eab380e9 parent 44725 d3bf0e33c98a child 44727 d45acd50a894
```     1.1 --- a/src/HOL/Transcendental.thy	Mon Sep 05 12:19:04 2011 -0700
1.2 +++ b/src/HOL/Transcendental.thy	Mon Sep 05 16:07:40 2011 -0700
1.3 @@ -54,7 +54,7 @@
1.4  x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
1.5
1.6  lemma powser_insidea:
1.7 -  fixes x z :: "'a::{real_normed_field,banach}"
1.8 +  fixes x z :: "'a::real_normed_field"
1.9    assumes 1: "summable (\<lambda>n. f n * x ^ n)"
1.10    assumes 2: "norm z < norm x"
1.11    shows "summable (\<lambda>n. norm (f n * z ^ n))"
1.12 @@ -65,7 +65,7 @@
1.13    hence "convergent (\<lambda>n. f n * x ^ n)"
1.14      by (rule convergentI)
1.15    hence "Cauchy (\<lambda>n. f n * x ^ n)"
1.16 -    by (simp add: Cauchy_convergent_iff)
1.17 +    by (rule convergent_Cauchy)
1.18    hence "Bseq (\<lambda>n. f n * x ^ n)"
1.19      by (rule Cauchy_Bseq)
1.20    then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
```