Have you seen the Khintchine-Kolmogorov Convergence Theorem?
It follows directly from that.
I found it online RIGHT out of Chow-Teicher.
I have all three editions of that book.
My copy from 78 is falling apart.
Teicher is my advisor's advisor and I published about 15 times In Chow's journal in Taiwan.
So, I use their book a lot.