# Math Help - Examples

1. ## Examples

Are there any examples that show that the pointwise limit of real-valued bounded functions need not be bounded?

E.g. $f(x) = \lim_{n \to \infty} f_{n}(x)$ is not bounded if each of the $f_{n}$'s are bounded?

2. Consider the sequence
$f_n=\begin{cases}
\sum_{k=0}^n x^n, &|\,x\,|<1\\
0, &|\,x\,|\geq 1.
\end{cases}$

Clearly, each $f_n$ is bounded, yet
$\lim_{n\to\infty}f_n(x)=\begin{cases}
\frac{1}{1-x}, &|\,x\,|<1\\
0, &|\,x\,|\geq 1.
\end{cases}$

is not.