real analysis - Prove that a conditionally convergent series has an infinity of positive terms and an infinity of negative terms.

The book I am using for my Advance Calculus course is Introduction to Analysis by Arthur Mattuck.

Prove that a conditionally convergent series has an infinity of positive terms and an infinity of negative terms.

This is my rough proof to this question. I was wondering if anybody can look over it and see if I made a mistake or if there is a simpler way of doing this problem. I want to thank you ahead of time it is greatly appreciated.So lets begin.

Proof: