Is clear to you that from the given we have
Therefore, we know that the series converges.
That implies .
I came across something used in another proof and I wanted to prove this simple fact for myself, although when I tried to pin is down with epsilons and the like, I found that I kept getting stuck! Considering sequences in . I would like to show that if then .
I would only like a hint with regards to how to do it, as I want to do it myself!
If I could prove a slightly weaker conclusion, that converges, then I can use the fact that convergent implies bounded and then I'd have something like
for suitably large n, then I could use the bound on to force to zero and therefor .
Am I over thinking this? Is there a simple proof? Some clever trick I have not seen?
Any help would be much appreciated, thanks