Let ${\left\{ {{c\_n}} \right\}\_{n \ge 1}}$, a bounded sequence of real numbers.

Let us define:
$$\begin{array}{l}
 {a\_n} = \inf \left\{ {{c\_k}:k \ge n} \right\} \\ 
 {b\_n} = \sup \left\{ {{c\_k}:k \ge n} \right\} \\ 
 \end{array}$$

$$\begin{array}{l}
 \lim \inf c\_n: = \mathop {\lim }\limits\_{n \to \infty } {a\_n} \\ 
 \lim \sup c\_n: = \mathop {\lim }\limits\_{n \to \infty } {b\_n} \\ 
 \end{array}$$

I don't understand very well the above sets definitions, and why it implies that:
$a\_n$ is the smallest partial limit and $b\_n$ is the greatest partial limit.