real analysis - Prove that $f(x)=x^2$ is uniformly continuous on any bounded interval.

May I please ask how to prove that $f(x)=x^2$ is uniformly continuous on any bounded interval? I know that there is a theorem saying that every continuous function on a compact set is uniformly continuous. But on the real line, "compactness"="closed and bounded". Why is that true that boundedness itself is sufficient for uniform continuity?