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