Let be a sequence of random variable with probability density of

Does converges almost surely?

Proof so far:

Claim: converges to 0 almost surely.

Let and fixed , I want to show that ,

that meansing proving that and

First the first one, we have:

since is decreasing monotonically. And that is the CDF of , which is:

as

Similar, the one will work as well. So is this it? Thanks.