--- a/src/HOL/Series.thy Mon Sep 05 12:19:04 2011 -0700
+++ b/src/HOL/Series.thy Mon Sep 05 16:07:40 2011 -0700
@@ -435,7 +435,7 @@
by (simp add: summable_def sums_def convergent_def)
lemma summable_LIMSEQ_zero:
- fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "summable f \<Longrightarrow> f ----> 0"
apply (drule summable_convergent_sumr_iff [THEN iffD1])
apply (drule convergent_Cauchy)
--- a/src/HOL/Transcendental.thy Mon Sep 05 12:19:04 2011 -0700
+++ b/src/HOL/Transcendental.thy Mon Sep 05 16:07:40 2011 -0700
@@ -54,7 +54,7 @@
x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
lemma powser_insidea:
- fixes x z :: "'a::{real_normed_field,banach}"
+ fixes x z :: "'a::real_normed_field"
assumes 1: "summable (\<lambda>n. f n * x ^ n)"
assumes 2: "norm z < norm x"
shows "summable (\<lambda>n. norm (f n * z ^ n))"
@@ -65,7 +65,7 @@
hence "convergent (\<lambda>n. f n * x ^ n)"
by (rule convergentI)
hence "Cauchy (\<lambda>n. f n * x ^ n)"
- by (simp add: Cauchy_convergent_iff)
+ by (rule convergent_Cauchy)
hence "Bseq (\<lambda>n. f n * x ^ n)"
by (rule Cauchy_Bseq)
then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"