# Proof of Proposition on Limits

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.

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@Steve,en.wikipedia.org/wiki/… , and use the fact that if there is supremum(infimum) it must be unique – pedja Sep 23 '11 at 4:38

HINT. Show that for each $n$, there exists some $y_n \in S$ such that $$\inf S \leq y_n \leq \inf S + \frac{1}{n}.$$ Show that the sequence $\{ y_n \}$ converges to $\inf S$. Can you take it from here?
As defined, the sequence $y_n$ is not guaranteed to be monotonic, but this can be easily fixed. Be sure to take care of this in the proof.