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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
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1 Answer

up vote 3 down vote accepted

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.

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I'll take all the help I can get thank you. I have the gist of it for the most part I believe –  Steve Sep 23 '11 at 2:42
    
That's what I figured when I was doing it on my own, thank you for all the help. I have shown that, again thank you –  Steve Sep 23 '11 at 3:00
    
@Steve I don't get you said. Are you able to complete it or do you want further help? –  Srivatsan Sep 23 '11 at 3:10
    
yes I was thank you –  Steve Sep 23 '11 at 3:35
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