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.
Best Answer
Your idea is good. You can pick $a_{k_1}$ such that $|a_{k_1}|<1/2$. Then pick $k_2>k_1$ so that $|a_{k_2}|<1/4$, and inductively pick ${k_n}>{k_{n-1}}$ such that $|a_{k_n}|<1/2^n$.