# An alternating series with non-decreasing subsequence

I have an alternating series $\displaystyle\sum_{n=0}^{\infty} (-1)^{n+1}a_n.$ I know that it is convergent. I have to find $a_n$ which is non-decreasing.

-
From your title it appears that you're looking for a subsequence but it's not clear from your text. The usual way to write a subsequence is ${a_{n_k}}$ ($k = 1, 2, 3...$). As stated, I think the claim that there's such a subsequence which you're trying to prove is false: suppose $a_n = 1/n$, then the series is convergent but every subsequence of ${a_n}$ is decreasing. –  Jonah Sinick Oct 29 '12 at 1:09
If the series is convergent, then $a_n\to 0$, and as $a_n\ge 0$, cannot have an infinite increasing subsequent. And a nondecreasing $a_n$ then must be quasiconstant $0$.