In the post (https://math.stackexchange.com/a/1729440/987565), the author states that $$\left\Vert \tilde{f}-f\right\Vert _{\sup}\leq\varepsilon$$ with $\tilde{f} = gf$, $f \in C_0(x)$, $g \in C_c(x)$, and $\text{supp } g$ is the open set $U = \{x \in X\ |\ |f(x)| > \epsilon/2\}$. The set $C_c(X)$ denotes the set of functions with that are compactly support on $X$.
It is clear that if $x \notin U$, we have $|(gf)(x) - f(x)| = |f(x)| < \epsilon/2$.
However, I am having issues verifying that $|f(x)||g(x) - 1| = |(gf)(x) - f(x)| < \epsilon$ when $x \in U$. I know that $f$ is bounded, but I am having trouble verifying how $|g(x) - 1|$ should help us establish the bound.
