Sequence of functions: Convergence For each of the following, give an example of a sequence of functions $f_n$ that converges to f
A. uniformly but not in the mean square sense.
B. in the mean square sense but pointwise nowhere.
I know that for part A the domain of the function cannot be bounded but other then that I'm not sure how to go about the problem

Hint A.
Set  $f_n(x) = g(n)1_{[0,n]}$ on $[0,\infty)$. 
Choose $g(n)$ such that $f_n(x) \to 0$ uniformly on $[0,\infty)$, but $\int_0^\infty |f_n(x)|^2\, dx \to L \neq 0$.
