Let be a sequence of absolutely continuous random variables such that has pdf
Show that the sequence converges in distribution. What happens when ?
to check for the convergence of a sequence of rv, you mustn't see the convergence of the pdf, but the convergence of the mgf, cdf, or characteristic function.
so for example, let's find the cdf :
by using the squeeze (or sandwich) theorem, we can show that
so the limiting cdf is
which is the cdf of a uniform distribution over
looks clear to you ?