Finding an example for Almost Surely convergence but not Mean Square convergence and vice versa.

Question

I've been trying to find an example for when a function converges almost surely but not mean square convergence and vice versa but I cant seem to find any easy examples on the web so I'm basically asking if any of you have some simple examples to help my understanding with them.

Answer 1

The sequence $f_n(x)=n1_{(0,\frac{1}{n})}$ converges to zero almost everywhere, but not in $L^2([0,1])$.