In this post, I’ll prove the Lebesgue Number Lemma and use it to prove that a continous function on a compact metric space is uniformly continuous.
Lemma 1 (Lebesgue Number Lemma). Let be a compact metric space and let be an open cover of . Then there exists a real number such that every open ball of radius is contained in some .
Proof: First, remark that if any refinement of the cover satisfies this property, then also satisfies this property; thus, since is compact, we may replace the cover by a finite cover by open balls .
Define on by
Then is continous and its support is . Let . Then is continous, and because covers . Since is compact, attains its minimum, which is ; we call it . Now, if , the statement means precisely that for some , , so the ball of radius around is contained in .
Theorem 2 Let be a continous function on the compact metric space . Then is uniformly continous.
Proof: Let . For each , let
Then is an open cover of . Let be a Lebesgue Number for the cover . Then, if are such that , there exists a such that ; therefore, since , we have