I'm working on the following question:
True or False (and explain why or give counterexample). Suppose $(f_n)$ uniformly converge to $f$. If each $f_n$ is bounded then $f$ is bounded.
I think this is true, but I'm having trouble showing it. In particular, suppose the $M_n$ is a bound for each $f_n$ then: $$|f_n-f| < \epsilon \\ -f_n - \epsilon < f < \epsilon + f_n \\ -M_n - \epsilon < f < \epsilon + M_n$$
There's nothing controlling the growth of the $M_n$'s though. If $f_n$ are uniformly converging, I think the bounds should be close too, but I'm having trouble arguing this. Thoughts?
This question is similar, but both answers skip the key detail that I'm hung up on.
Your bound is correct for any given $n$ (depending only on the choice of $\varepsilon$). So just choose $\varepsilon =1$, find a suitable $n$ and fix that $\Rightarrow f$ is bounded.