Prove that a series $\sum^{\infty}_{n=1} (a_n - a_{n+1})$ converges iff the sequence $\{ a_n \}$ converges.
Since $\sum^{k}_{n=1} (a_n - a_{n+1}) = a_1 - a_{k+1}$, the sequence $a_n$ is obviously convergent, but how can I prove it real analytically(formally)?