I need to prove that the sequence {$\displaystyle a_n$} converges if and only if $\displaystyle \sum _{n=1}^{\infty} (a_{n+1} - a_n)$ converges. I think I've managed to prove that if the series converges then the sequence converges, but I'm having trouble with the other direction. I have a feeling I'm missing something really obvious, but any help would be appreciated.