Let $(X,\mathfrak{A},\mu)$ be a measurable space, $f_n,n\geqslant 1$ measurable and bounded functions with $f_n\to f$ uniformly. Would like to know if then $f$ is measurable and bounded, too.
I think the measurability of $f$ follows from the fact that - if $f_n, n\geqslant 1$ is measurable - then $\lim_n f_n$ is measurable too, if this limit exists. Here it exists, anyway the task sounds like this.
Now I would like to prove that $f$ is bounded, i.e. I have to find a $M\geqslant 0$, so that $\lvert f(x)\rvert\leqslant M$ for all $x\in X$. Do not know if my proof is correct.
$$ \lvert f(x)\rvert=\sup_n\lvert f_n(x)\rvert $$