# 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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If this is homework, please add the [homework] tag. Also, can you tell us what you have tried? Where are you stuck? –  Srivatsan Sep 23 '11 at 1:54
Oh sorry I will add the homework tag, also I am more or less stuck in where to begin as I feel that it is an obvious result but I can't see how to get there formally. From previous results of a monotone decreasing having the limit of inf and monotone increasing the limit of sup, I would think it should be, but again I don't really know. –  Steve Sep 23 '11 at 1:59
@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.