TITLE: Proof that a sequence that converges to zero has a subsequence whose series is convergent
QUESTION [8 upvotes]: How would I go about proving that if $a_n$ is a real sequence such that $\lim_{n\to\infty}|a_n|=0$, then there exists a subsequence of $a_n$, which we call $a_{n_k}$, such that $\sum_{k=1}^\infty a_{n_k}$ is convergent.
I think that I can choose terms $a_{n_k}$ such that they are terms of a geometric series, so that means that it will converge, but I don't know how to formally state this.

REPLY [4 votes]: Start with a series you know is convergent. A geometric series will work as you have guessed. I'll use the series $\sum_{n=1}^\infty \frac{1}{n^2}$ for my example. Choose each $a_{n_{k}}$ such that $\left|a_{n_{k}}\right|\leq \frac{1}{n_k^2}$ for some integer $n_k$. You will need to prove that you can find infinitely many of these $a_{n_{k}}$. Once you know that, then $$\sum_{n=1}^\infty \left|a_{n_{k}}\right|\leq \sum_{k=1}^\infty \frac{1}{n_k^2} \leq \sum_{n=1}^\infty \frac{1}{n^2}$$ Hence, $$\sum_{n=1}^\infty a_{n_{k}}$$ is an absolutely convergent series.