Claim:The series converges.

By definition:

The series converges if and only if for every , there exists some such that for all with , we have .

(Attempted) proof:

Let be arbitrary but fixed, and let . Then the sequence is convergent, and it necessarily follows that is Cauchy. That is, for all , there exists some such that for all , we have .

But if , we have

which shows, directly from the definition, that the series is convergent.

Is this satisfactory? Or is there a more simplistic approach that I should be taking?