I need help proving the following proposition. Thank you for any help you can give me.
Let $S \subset \mathbb R$ be a nonempty bounded set. Then there exist monotone sequences $\{ x_n \}$ and $\{ y_n \}$ such that $x_n, y_n \in S$ and $ \sup S = \lim_{n \to \infty} x_n \ \ \ \ \ \text{ and } \ \ \ \ \ \inf S = \lim_{n \to \infty} y_n .$
Thank you again.