Basically, prove the Extreme Value Theorem.
"If f is a continuous function over the interval [a,b] then f reaches a max and a min on that interval."
In this case they're more like definitions and things I have proved so far.
Intervals are connected.
The only connected subsets of ℝ are intervals.
A subset S of ℝ is compact iff every infinite subset of S has a limit point in S.
Closed intervals are compact
A subset of ℝ is closed iff it contains all its limit points
The Attempt at a Solution
En route to this problem I've shown that the image of a compact set by a continuous function is compact and that the image of a connected function by a continuous function is connected. Because an [a,b] is connected its image by f has to be an interval as well, by the second definition above. This image is compact also since [a,b] is compact. And we know that intervals are bounded.
All in all, I know that the image of [a,b] by f is a compact, bounded interval. I'm trying to use the definition of compact to show that the image has to also be closed, since my feeling is that will help me prove the EVT.
Right now I'm trying contradiction, supposing the image is compact but has a limit point outside of it. I'm not sure where to go from there in terms of arriving at a contraction.
Does it seem like I am on at all on the right track?
Thanks for your help.