Proving non-uniform-continuity for f,g when g(x) > f(x) for all x

Question

Suppose $f, g$ are continuous functions on $\mathbb{R}$, s.t $f$ is not uniformly continuous on $\mathbb{R}$ and $g(x) > f(x)$ for all x. Is g not uniformly continuous?

This seems true, and I'm trying to prove this with an 'epsilon delta' proof, without much success.

I tried to negate the uniform continuity condition, which means I need to find x,y that will make $g(x)$ and $g(y)$ further apart than $f(x)$ and $f(y)$ are (so the distance will be bigger than/equal to epsilon).

Any help at all is appreciated :) Preferably a hint

Answer 1

This is not true. Consider $f(x) = \sin(x^2)$ and $g(x) = 2$.