Bound for the difference between two functions

Question

Consider $f_1, f_2, g_1, g_2$ four continuous functions defined on the real line.

I know that for every $x \in \mathbb{R}$

$$0 \leq |f_1(x)| \leq |g_1(x)|, \quad 0 \leq |f_2(x)| \leq |g_2(x)|$$

If I manage to prove that $g_1$ and $g_2$ are such that

$$| |g_1(x)| - |g_2(x)| | = 0$$

does it implies that

$$| f_1(x) - f_2(x) | = 0 ?$$

Answer 1

No. One simple counterexample is $g_1=g_2 \equiv 42$ and $f_1 = 0$ and $f_2 \equiv 1$