Let $(f_n)$ be a sequence that converges uniformly to some $f$.
I want to show that the sequence of absolute values $(|f_n|))$ converges uniformly to $|f|$.
Proof:
Uniform convergence of $f_n\to f$ is equivalent to convergence with regards to the supremum norm.
This means $\displaystyle \lim_{n\to\infty} \|f_n-f\|_\infty =0$.
Now to show $\displaystyle \lim_{n\to\infty} \||f_n|-|f|\|=0$, we use the reversed triangle inequality.
$\displaystyle \lim_{n\to\infty} \||f_n|-|f|\|_\infty\leq \lim_{n\to\infty} \|f_n-f\|_\infty=0$.
Is this proof correct? Thank you.