Let S ⊂ R be a nonempty bounded set. Then there exist monotone sequences ${x_n}$ and ${y_n}$ such that $x_n$, $y_n$ ∈ $S$

Let S ⊂ R be a nonempty bounded set. Then there exist monotone sequences ${x_n}$ and ${y_n}$ such that $x_n$, $y_n$ ∈ $S$ and $\sup S = \lim_{n→∞} x_n$ and $\inf S = \lim_{n→∞} y_n$

How can I prove this ..?