how do I proof the following lemma:
independent r.v. with and .
For the first problem, compute .
If we assume that the sequence converges in probability then the limit when will be , and it show that the limit of the sequence is (the converse will follow from the previous computation).
For the second problem, use Borel-Cantelli lemma.
It's an interesting exercise, since we can easily give an example of a sequence of random variables which converges in probability but not almost everywhere.