I wonder if the following statement is true and the proof is right:
[Statement] Let $X \subset \mathbb{R}^n$ be a compact set and $\overline{int(X)} = X$. Given $\epsilon > 0 $, if there exists a function $g_{\epsilon}: X \rightarrow \mathbb{R}$ such that $f(x) \leq g_{\epsilon}(x) \leq f(x) + h(\epsilon), \forall x \in int(X)$ where $h(\epsilon) \rightarrow 0+$ as $\epsilon \rightarrow 0+$ holds true for a continuous function $f$, then $g_{\epsilon} \rightarrow f$ uniformly on $X$.
[Proof] Since $f(x) \leq g_{\epsilon}(x) \leq f(x) + h(\epsilon), \forall x \in int(X)$, $g_{\epsilon}$ convergence pointwise to $f$ as $\epsilon \rightarrow 0+$. Further, $f$ is continuous. By the second theorem of Ascoli (see here), $g_{\epsilon}$ converges uniformly to $f$ on $\overline{int(X)} = X$.
EDIT: $g_{\epsilon}$ is continuous.