Are there any examples that show that the pointwise limit of real-valued bounded functions need not be bounded?
E.g. $\displaystyle f(x) = \lim_{n \to \infty} f_{n}(x) $ is not bounded if each of the $\displaystyle f_{n} $'s are bounded?
Consider the sequence
$\displaystyle f_n=\begin{cases}
\sum_{k=0}^n x^n, &|\,x\,|<1\\
0, &|\,x\,|\geq 1.
\end{cases}$
Clearly, each $\displaystyle f_n$ is bounded, yet
$\displaystyle \lim_{n\to\infty}f_n(x)=\begin{cases}
\frac{1}{1-x}, &|\,x\,|<1\\
0, &|\,x\,|\geq 1.
\end{cases}$
is not.