1. ## Proof of convergence

I need to prove that the sequence { $a_n$} converges if and only if $\sum _{n=1}^{\infty} (a_{n+1} - a_n)$ 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.

2. Originally Posted by spoon737
I need to prove that the sequence { $a_n$} converges if and only if $\sum _{n=1}^{\infty} (a_{n+1} - a_n)$ 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.
If $\{ a_n \}$ converges it means $|a_n - a_m| < \epsilon$ by using the Cauchy sequence. But then it means $\left| \sum_{k=m}^{n-1} a_{k+1} - a_k\right| = |a_n - a_m| < \epsilon$. Thus, the series satisfies the Cauchy condition for convergence.

3. Thanks for the help, but we haven't learned the Cauchy convergence criterion for series yet, so I don't think I can use it. Is there another way I can approach this?

4. Originally Posted by spoon737
Thanks for the help, but we haven't learned the Cauchy convergence criterion for series yet, so I don't think I can use it. Is there another way I can approach this?
Well it is the same thing as for sequences. If $s_n$ is the sequence of partial sums then $|s_n - s_m|$ is that finite sum I wrote up there. All it is, is the ordinary Cauchy sequence but applied to a sequence of partial sums.

5. Okay, that makes sense. Thanks again.