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Prove: If f : U → R is continuous on U, and E ⊂ U is closed and bounded, then
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.
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.