First center the random variables and then use Kolmogorov's Three Series and then Kroncker's lemma.
You can actually quote the literature.
I proved this type of result in my dissertation for non-independent stochastically dominated rvs a long time ago.
I found the exact result (exact is a pun by the way).
See Corollary 3 on page 128 from Chow and Teicher, which follows from Theorem 3.
You should avoid the three series theorem, because we don't even want to mention having a second moment here.