is uniform convergent sequence leads to bounded function?
Your proof is almost correct, but it seems quite awkward at points and doesn't state the correct conditions on variables (you have incorrectly quantified $n$ and $N$, for example). What I think you mean is:
By uniform convergence, there exists an $N$ such that $|f_n(x) - f(x)| \le 1$ for all $n \ge N$. Then $$|f(x)| \le |f_N(x)| + 1 \le M_N + 1$$
Note that we can explicitly state the bound in terms of $M_N$ and don't need to take a maximum to bound $f$ itself.