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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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    @Steve,http://en.wikipedia.org/wiki/Monotone_convergence_theorem#Convergence_of_a_monotone_sequence_of_real_numbers , and use the fact that if there is supremum(infimum) it must be unique2011-09-23

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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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    @Steve I don't get you said. Are you able to complete it or do you want further help?2011-09-23