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0568 Q62 proof

Posted: Tue Mar 20, 2012 1:32 am
by yoyostein
Hi, could anyone give a proof of why Q62 II is true?

II. If every continuous real-valued function defined on K is bounded, then K is compact.
(K nonempty subset of R^n)

Thank you very much!

Re: 0568 Q62 proof

Posted: Tue Mar 20, 2012 2:03 am
by blitzer6266
First of all, in euclidean space, compact=closed and bounded

Suppose K is not compact, then either it is not closed or not bounded

Not bounded- consider the function ||x|| which will not be bounded
Not closed-there exists a limit point A such that x_n goes to A but A is not in K
Then you can look at the function 1/||x - A||. This will be continuous on K but unbounded since it goes to infinity as x goes to A

There are probably better proofs but this is at least the picture that should come to mind.

Re: 0568 Q62 proof

Posted: Tue Mar 20, 2012 7:16 am
by yoyostein
thanks!

this is crystal clear explanation