If $X=(x_n)$ is a bounded sequence in $\mathbb{R}$ show that there is
a subsequence of $X$ which converges to $\lim \text{inf} (x_n)$.
My attempt:
Let $\lim \text{inf} (x_n) =x$. Then by definition, there are infintely many $x_n$ such that $$x-\epsilon <x_n<x+\epsilon.$$ Which implies that $\exists$ $x_{n_k}$ such that
$x_{n_k}\in \{a-\epsilon,a+\epsilon\}= a-\epsilon <x_n<a+\epsilon$…
But now how can I get my subsequence to converge to $\lim \text{inf} (x_n)$?
Best Answer
Choose $\varepsilon_n = \frac{1}{n}$ and use your method to extract a single $x_n$ for each $\varepsilon_n$. Because there are infinitely many $x_n$ to choose from for each $\varepsilon_n$, we can make it so that we choose in order so that it actually is a subsequence.