(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

a) Let U be a closed subset of the reals with an upper bound. You know that U has a supremum, say z. Prove that z is an element of U.

b) Suppose U is a closed subset of the real numbers with an upper and lower bound. Prove that U has a maximum and minimum.

3. The attempt at a solution

I was able to come up with a proof for b) by simply finding the maximum and minimum of U however I feel like the proof is flawed.

b) Since U is a closed and bounded subset of R, we can write U as a finite collection of closed intervals of R.

[tex] U = \cup^{n}_{1} I_{i}[/tex] where [tex] I_{i} = [a_{i},b_{i}]. [/tex]

Hence for each [tex]I_{i}[/tex],

[tex]max(I_{i}) = b_{i}[/tex]

and

[tex]min(I_{i}) = a_{i}[/tex].

Then consider the set

[tex]S = (a_{i})\cup(b_{i}) [/tex] for i = 1,...,n.

Then [tex]max(s) = b_{max} [/tex] and [tex] min(S) = a_{min}[/tex].

So max(U) and min(U) both exist and are equal to bmax and amax respectively.

a) I wasn't sure how to do this one. I tried using some inequalities and defining the maximum of U but couldn't come up with anything useful.

Any help is greatly appreciated.

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# Homework Help: Analysis: Closed sets and extrema

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