Is it true that $\left|\max|f(x)| - \max|g(x)|\right| \leq \max|f(x) - g(x)|$?

Question

Is it true that $\left|\max|f(x)| - \max|g(x)|\right| \leq \max\left||f(x)| - |g(x)| \right| \leq \max\left|f(x) - g(x)\right|$ for $x \in A$ (assuming it is possible to attain any maxima desired)? Proof?

Answer 1

$L^{\infty}(\mu)$ is a normed space, in particular: $\big|\|f\|_{L^{\infty}}-\|g\|_{L^{\infty}}\big|\leq\|f-g\|_{L^{\infty}}$.