Let . The sequence is decreasing and positive so the limit exists. We have
and by the substitution we get .
For all we have the inequality hence .
Now we try to compute . By putting we get
. We can see that the two integrals are the same (by the substitution ). So we only have to compute the first one.
We have . Then and we conclude the limit we were looking for is .
It is an immediate consequence of the dominated convergence theorem that the limit is 0.
The integrand is non-negative and converges monotonically to almost everywhere, so you may switch the limit and the integral.