Section Isequence.
(*********)
Variable An : nat -> R.
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Definition Pser (x l:R) : Prop := infinite_sum (fun n:nat => An n * x ^ n) l.
End Isequence.
intros f' k l cvfk eps ep; destruct (cvfk eps ep) as [N Pn].
exists (N + k)%nat; intros n nN; assert (tmp: (n = (n - k) + k)%nat).
rewrite Nat.sub_add;[ | apply le_trans with (N + k)%nat]; auto with arith.
rewrite tmp; apply Pn; apply Nat.le_add_le_sub_r; assumption.
Qed.
intros f' k l cvf eps ep; destruct (cvf eps ep) as [N Pn].
exists N; intros n nN; apply Pn; auto with arith.
Qed.
(* Growing property is preserved after shifting the indices (one way only) *)

Lemma Un_growing_shift :
forall k un, Un_growing un -> Un_growing (fun n => un (n + k)%nat).