How to prove the limit inferior of a bounded squence exists?

Question

Let $\{a_k\}$ be a bounded sequence of real numbers. Define a sequence $b_k =$inf$ \{a_l|l\ge k \}$ for $k\ge1$. Prove that $\lim_{k\to \infty }$$b_k$ exists.

I am proving that the limit inferior exists, but I don't know where to start. How can we show the limit exists?

Answer 1

HINT: Show that if $k\le\ell$, then $b_k\le b_\ell$. Thus, the sequence $\langle b_k:k\in\Bbb N\rangle$ is a bounded, monotone non-decreasing sequence. Therefore ... ?