real analysis - Prove that if $a_n$ is a non-negative sequence, lim $a_n$ = 0 $implies$ lim $sqrt{a_n}$ =0

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

Prove that if $a_n$ is a non-negative sequence, lim $a_n$ = 0 $\implies$ lim $\sqrt{a_n}$ =0

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: