This is an analysis question I got stuck on:

I decided to make $\displaystyle a_N$ my sequence. I need to show that this is bounded if $\displaystyle a_{N+n}$ is bounded.Prove that if a sequence is eventually bounded then it is bounded.

So my first step was:

$\displaystyle |a_{n+N}|<\epsilon$

I was not sure where to go from here. Is there a way to separate $\displaystyle a_{n+N}$? I was also thinking that changing my epsilon to an expression would help but I can't see what to change it to.

Help would be very much appreciated.