- #1
iopmar06
- 14
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Homework Statement
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.
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.