Have you seen the Khintchine-Kolmogorov Convergence Theorem?

It follows directly from that.

I found it online RIGHT out of Chow-Teicher.

http://books.google.com/books?id=mYn...heorem&f=false

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.

http://en.wikipedia.org/wiki/Yuan-Shih_Chow

(Academia Sinica)

So, I use their book a lot.