Non-uniform convergence in a compact metric space $K$ is a compact metric space and we are given a pair of continuous functions $f$ and $g$ : $K \rightarrow \mathbb{R}$ such that $f(x)$ is greater than $g(x)$ for all $x \in X$; Prove that there exists an $\epsilon$ such that $f(x)$ is greater than $g(x) + \epsilon$ for all $x \in X$
I was going to construct two sequences of continuous functions $f_n$ and $g_n$ and somehow show that they don't converge uniformly to the same point but then I got lost
Thanks for your help in advance!

Let $h(x):=f(x)-g(x)$ for all $x\in K$. Then $h$ is continuous and positive. Since we work on a compact space, the minimal value of $h$ is reached at say $x_0\in K$. So we can put $\varepsilon:=\frac{h(x_0)}2>0$: $f(x)\geq g(x)+h(x_0)>g(x)+\varepsilon$.
