Let a sequence $(x_n)_{n\geq 1}$ be a bounded sequence of real numbers. Define $\mathbf S$ as the adherent set of the sequence. Prove that $\lim \inf\;(x_n)=\inf(S)$.
An adherent set $\mathbf S$ is a set that consists of the limiting points of subsequences of $x_n$.
I have made several attempts but I do not how to proceed.
I believe the bounded property of $x_n$ in combination with the Bolzano-Weierstrass theorem probably would be useful. Thanks in advance.