Hi, could anyone give a proof of why Q62 II is true?
II. If every continuous realvalued function defined on K is bounded, then K is compact.
(K nonempty subset of R^n)
Thank you very much!
0568 Q62 proof

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 Joined: Sun Apr 04, 2010 1:08 pm
Re: 0568 Q62 proof
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 closedthere 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.
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 closedthere 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
thanks!
this is crystal clear explanation
this is crystal clear explanation