Check my probability proof?

Hello all. I'm not confident in my methodology for this, so confirmation that this is okay would be helpful.

**Problem: **Let be iid random variables with . Let . Show that .

**Solution:** We must show that , which is equivalent to . By the weak law of large numbers, we know , and hence holds if and only if is weakly uniformly integrable.

First we show is uniformly integrable.

But because the rvs are identically distributed, this is equivalent to

which is true. Next we show is weakly uniformly integrable. Fix and using uniform integrability of the get such that if then . Fix with . Then

and we are done.

Re: Check my probability proof?

What you did is correct. Just add the fact that is bounded.