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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)?

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    $\begingroup$ What you did is also correct $\endgroup$ Sep 17, 2020 at 12:09

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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

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