Let $\displaystyle X_1, X_2, .. $ be a sequence of absolutely continuous random variables such that $\displaystyle X_n $ has pdf

$\displaystyle f_n(x) = \begin{cases} 1 - \cos (2 \pi n x) & x \in [0,1], \\ 0 & otherwise \end{cases}$.

Show that the sequence converges in distribution. What happens when $\displaystyle f_n \rightarrow \infty $?