Question: Prove that every polynomial bounded on the compact interval [a,b] is of bounded variation
My proof: Let $f(x)$ be a bounded polynomial of n degree. Then $f '(x)$ is also a polynomial bounded on [a,b] and also $f(x)$ is continuous on [a,b] since $f(x)$ is a polynomial. Thus $f$ is of bounded variation on [a,b].
I would appreciate if someone could point out any mistakes in my proof or if more justifications are needed. Thanks