equal
deleted
inserted
replaced
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 |