# Pointwise Convergence

1. Apr 9, 2013

### Arkuski

Give an example of a sequence $\{ f_n\}$ of continuous functions defined on [0,1] such that $\{ f_n\}$ converges pointwise to the zero function on [0,1], but the sequence $\{ \int^{1}_{0} f_n\}$ is unbounded.

I'm pretty lost on this one.

2. Apr 9, 2013

### jbunniii

Hint: Try coming up with a sequence of functions such that $f_n(x)$ is only nonzero on the interval $(0, 1/n)$. You will need to make them grow "taller" and "narrower" as $n$ increases.