Let $h : (0, +\infty) \to \mathbb{R}$ be the function defined by $$h(x) = \cos(2x) + x\sin(1/x).$$
Prove that $h$ is uniformly continuous on $(0, +\infty)$.
My attempt: $h$ is uniformly continuous on $(0, +\infty)$ if for each $\epsilon>0$, there's a $\delta>0$ such that $|h(x)-h(y)|<\epsilon$ for all $|x-y|<\delta$, $x,y\in (0, +\infty)$.