Hello I want to prove that $f(x)=x^n$ is uniformly continuous on any bounded subset of $\mathbb{R}$. I'm wondering if my proof is correct, and I'm primarily wondering if my choice of $\delta$ is correct. Thanks!
Proof.
Let $A$ be a bounded subset of $\mathbb{R}$. Then there exists $M > 0$ such that $|a| \le M$ for every $a \in A$.
Let $\epsilon > 0$ be given. Now choose $\delta = \frac{\epsilon}{nM^{n-1}}$. Then for every $x,y \in A$ if $|x-y| < \delta$ then
$$|x^{n} - y^{n}| = |x-y||x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1}| \\ \le |x-y|nM^{n-1} < \frac{\epsilon}{nM^{n-1}} nM^{n-1} = \epsilon.$$
Thus f is uniformly continuous on A.