Prove that a series $\sum^{\infty}_{n=1} (a_n - a_{n+1})$ converges iff the sequence $\{ a_n \}$ converges.

Question

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)?

Answer 1

For some real $\alpha$ $$\lim_{n\to∞}|a_n|<\alpha$$ $$\implies\lim_{n\to∞}|a_1-a_n|\leq |a_1|+\alpha$$

That is a way how you can present your intuition mathematically