I am required to prove that for any non-negative sequence of real numbers say $(s_n)$ we have $\lim \sqrt{s_n} = 0$ whenever $\lim s_n = 0$.
The following is my attempt. I would appreciate if some one could critique the following argument and determine whether is correct or not?
Proof. Assume that $\lim s_n = 0$. Let $\epsilon>0$ from hypothesis we may choose a $N\in\mathbf{R}$ such that given any $n\in\mathbf{Z^+}$ we have $|s_n|<\epsilon^2$ whenever $n>N$ but since $s_n\ge0$ the aforementioned inequalty is equivalent to $|\sqrt{s_n}|^2<\epsilon^2$ then taking square root yields $|\sqrt{s_n}|<\epsilon$. Since $\epsilon>0$ was arbitrary it follows that $\lim \sqrt{s_n} = 0$.
$\blacksquare$
You proved correctly that$$\lim_{n\in\mathbb N}s_n=0\implies\lim_{n\in\mathbb N}\sqrt{s_n}=0.$$You didn't prove the reverse implication, but the proof is almost identical.