I found the following statement in the internet and I have a conceptual question, please: Let $f_n$ be a sequence of functions such that $f_n\to f$ uniformly as $n\to\infty$ and each $f_n$ bounded. Show that $f_n$ is uniformly bounded, i.e. $\exists M>0 \ \forall x\in\mathbf{R} \ \forall n\in\mathbf{N}: |f_n(x)|\le M$.
So, the thing that I don't understand, is why the hypithesis that $f_n\to f$ uniformly is a necessary condition to get the uniform boundness? Why the pointwise convergence is not enough? I was thinking on some examples but I didn't find anything. If someone could provide some, I would really appreciate it. The key thing in this statement, I think, that with 2 conditions ($f_n\to f$ uniformly and $f_n$ bounded) we are sure that $f_n$ limit doesn't "explode", is it correct?
If $f_n(x)=n$ for $x>n$ and $0$ for $x \leq n$ then $f_n$ is bounded by $n$ for each $n$, $f_n \to 0$ point-wise but $f_n$ is not uniformly bounded.