What have you done so far? Some basic hints, Markov's inequality will be useful and also splitting the expectation.
I was trying with Markov's inequality earlier but didn't get anywhere. I think I've managed to show one inclusion now. I don't see how to split the expectation either which is maybe why I'm stuck on the other.
So I have expectation >> convergence since for small .
You are supposed to show that which is pretty similar to what you did if you notice that .
For the converse try splitting as follows
You can bound the RHS term by epsilon (as I indicated above), the second term you can split further by using the fact that then use convergence in probability.