real analysis - Prove that if $epsilon > 0$ is given, then $frac{n}{n+2}$ ${approx_epsilon}$ 1, for $n$ $gg$1.

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

Prove that if $\epsilon > 0$ is given, then $\frac{n}{n+2}$ ${\approx_\epsilon}$ 1, for $n$ $\gg$1.

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: