Prove: If f : U → R is continuous on U, and E ⊂ U is closed and bounded, then(adsbygoogle = window.adsbygoogle || []).push({});

f attains an absolute minimum and maximum on E.

I have no idea even where to start on this. Intuitively it's so obvious that i don't know what to do. The definitions given by the teachers that I have to work with are as follows:

A set F ⊆ Rn is closed if, for every convergent sequence {xi}(from i=1 to infinity) ⊆ F,

we have limxn(as n goes to infinity)⊆F

(in other words it contains its limit points)

A bounded set is one for which there exists r such that the set is contained in Nr(0).

(in other words some ball around the origin of any size contains the set.)

F continuous on u means for all c in F, lim(as x approaches c) exists and equals F(c).

A compact set is one which is closed and bounded, so E in this proof is compact.

I know that what needs to be shown is that there exists xm and xM such that:

f(xm)<=f(x)<=f(xM) for all x in E.

any advice? hints? etc.. on how to start this? I suspect that the mean value theorem might have something to do with it, but I'm not sure how to incorporate it. Thanks for any help.

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# Mean value theorem

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