TITLE: $\{x_n\}$ be a bounded sequence of real number we need to show
QUESTION [8 upvotes]: Let $\{x_n\}$ be a bounded sequence of real numbers. We need to show that there exist a  real number $\alpha$ and positive integers $n_1,n_2,\dots$ such that $n_1<n_2<\dots$ and $\sum_{k}|x_{n_k}-\alpha|<\infty$, please hint!

REPLY [7 votes]: By Bolzano-Weierstrass there exists a limit point $\alpha$. Then for each $k$ select $n_k>n_{k-1}$ with $|\alpha-x_{n_k}|<2^{-k}$.