Consider an interval from a to b that we want to show is closed. Given that the interval is bounded and non empty, we know it must contain a supremum. Now we want to show that there exists a max, which will necessarily be the supremum prove that half of the interval is closed i.e. (a,b].
Anyone know how to do so?
Prove. Let F[a,b]-->R be a continuous function on a compact interval then the set of values of {f(x)|a<=x<=b} has a max and a min.
I know there exists a supremum and infinimum, now I need to show there exists a maximum -- which would be one of my endpoints. Then, repeat the process for a minimum.
Can anyway please generalize this property for me? My teacher hinted that it would be on our midterm, but not in the context of continuity..? He just said he would give us a set and we would know that a supremum existed, and that we would have to show a max existed. Ideas anyone?
Thanks.
Just to clear things up this makes sense right?
Given a bounded non-empty set then a by the completeness property.
Now take a sequence such that Then by Bolzano Weierstrass there exists a subsequence But converges so by the subsequence theorem we have
Thanks everyone.