src/HOL/Hyperreal/Series.thy
changeset 27108 e447b3107696
parent 23441 ee218296d635
equal deleted inserted replaced
27107:4a7415c67063 27108:e447b3107696
   539 apply (rule_tac y = "c * norm (f (N + n))" in order_trans)
   539 apply (rule_tac y = "c * norm (f (N + n))" in order_trans)
   540 apply (auto intro: mult_right_mono simp add: summable_def)
   540 apply (auto intro: mult_right_mono simp add: summable_def)
   541 apply (simp add: mult_ac)
   541 apply (simp add: mult_ac)
   542 apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
   542 apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
   543 apply (rule sums_divide) 
   543 apply (rule sums_divide) 
   544 apply (rule sums_mult) 
   544 apply (rule sums_mult)
   545 apply (auto intro!: geometric_sums)
   545 apply (auto intro!: geometric_sums)
   546 done
   546 done
   547 
   547 
   548 subsection {* Cauchy Product Formula *}
   548 subsection {* Cauchy Product Formula *}
   549 
   549