For case I and case II, can you please explain how professor came to what epsilon was equal to?

https://imgur.com/a/BU09F6d

This may not help you at all. But I refuse to jump into someone else's proof.

We are given that $(\forall n)[S_n\ge 0]~\&~ S_n\to S$

1) We are to prove that $S\ge 0$ So let's suppose that $S<0$

Let $\varepsilon = \frac{{ - S}}{2}$. By convergence $(\exists N\in\mathbb{Z}^+)[n\ge N \Rightarrow |S - {S_n}| < \varepsilon$

Thus we have $\begin{gathered} \left| {{S_n} - S} \right| < \varepsilon \hfill \\ - \varepsilon < {S_n} - S < \varepsilon \hfill \\

{S_n} < \varepsilon + S < \frac{{ - S}}{2} + S = \frac{S}{2} < 0 \hfill \\ \end{gathered}$ But $S_N\ge 0$ so there is a contradiction.

Hence $S\ge 0$ as you were asked to prove for 1).

Now can you construct a proof for $\sqrt{S_n}\to\sqrt{S}~?$