Problem: Is function unifomly continuous?

Question

Is sin ( 1/x ) uniformly continuous on set (1, + infinity) ? I tried to prove that it is Lipschitz continuous but I got stuck.

Answer 1

Certainly: it's differentiable with bounded first derivative. First property of Lipschitz continuous functions as listed on Wikipedia:

An everywhere differentiable function $g : \mathbb{R} \to \mathbb{R}$ is Lipschitz continuous (with $K = \sup |g'(x)|$) if and only if it has bounded first derivative; one direction follows from the mean value theorem."