real analysis - By choosing one word within each parentheses,four statements can be made.

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

By choosing one word within each parentheses, four statements can be made from the following. Label each true, or false (with counterexample).

If lim $a_n$ is (positive, non-negative), then for large n the individuals term $a_n$ are (positive, non-negative).

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: