Let $(f_n)$ be a sequence of bounded functions on a set $S$, and suppose $f_n \to f$ uniformly on $S$. Prove $f$ is bounded on $S$.
By uniform convergence $\forall \epsilon \ \exists N>0 : \forall x \in S,n>N$ $$|f_n(x) - f(x)| < \epsilon$$ By the reverse triangle inequality we can obtain $$|f(x)| \leq |f_n(x)|+|f_n(x)-f(x)| < |f_n(x)| + \epsilon$$
Since $\forall n, |f_n(x)| < M_n$
$$|f(x)| < \epsilon' = \epsilon + M_n$$
and $f$ is bounded on $S$.
Why do I need to fix $\epsilon$ if it is arbitrary?