Finite fourth moment indeed suggests to apply Chebychev inequality, expand the square, and pray for not losing a few terms during the computation.

Actually, you don't really need that fourth moment; second moment suffices. Together with weak law of large numbers (wLLN).

Remember . Likewise, and, after summation (split the sum into four sums), the terms with "bars" factorize and the last three sums are the same, hence .

This way, we may split into two simple terms. If both of them converge to 0 in probability, then so does their sum (you may need to prove that).

First term is . Just apply wLLN.

Second term is . Apply wLLN and the fact that the product of two sequences converging to 0 in proba converges to 0 as well (you may need to prove this).

In fact, applying the strong law of large numbers, you even get an almost-sure convergence and this eliminates all problems of justifications (it is obvious (do you understand why?) that the sum or the product of two sequences converging a.s. to 0 converges a.s. to 0 as well).