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# Heine-Borel Theorem

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Let f:[a,b]--> R be continuous. Prove that the set f([a,b]) has both a least upper bound M and a greatest lower bound m and that there are points u,v in [a,b] such that f(u)=M, f(v)=m.

##### Solution Summary

Extreme Value Theorem from calculus is infused in this solution.

##### Solution Preview

Let be continuous. Prove that the set has both a least upper bound M and a greatest lower bound m and that there are points u, v, in such that and

Proof: Note that this is the Extreme Value Theorem from calculus. To prove it, we can make use of two theorems from topology.

Theorem 1. Heine-Borel Theorem. A set is compact if and only if E is closed and bounded.

Theorem 2. Let ...

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