Possible Error in Elementary Analysis by Ross I am having some serious trouble understanding example 5 from chapter 8 of Elementary Analysis by Ross. The example is to do with proving sequence are convergent, and here is the definition of convergence that the example uses:

A sequence ($s_n$) is said to converge to the real number $s$ provided that for each $\epsilon>0$ there exists a number $N$ such that $n>N$ implies $|s_n-s|<\epsilon$.

Now, the example is as follows:

Let ($s_n$) be a sequence of nonnegative real numbers and suppose $s=\text{lim}s_n$. Note that $s \geq 0$. Prove that lim$\sqrt{s_n}=$lim$\sqrt{s}$.

Next, the author divides the work into two cases: when $s=0$ and when $s>0$. The case in which $s=0$ is left as an exercise and just by inspection appears trivial, but my question is regarding when $s>0$.
He shows that for $\epsilon>0$ we must prove $\exists N \in \mathbb{N}$ such that $n>N$ implies $|\sqrt{s_n}-\sqrt{s}|<\epsilon$
For $s>0$, we have that
$$|\sqrt{s_n}-\sqrt{s}|=\frac{|\sqrt{s_n}-\sqrt{s}||\sqrt{s_n}+\sqrt{s}|}{|\sqrt{s_n}+\sqrt{s}|}=\frac{|s_n-s|}{\sqrt{s_n}+\sqrt{s}}\leq\frac{|s_n-s|}{\sqrt{s}},$$
So we will select $N$ so that $|s_n-s|<\sqrt{s}\epsilon$ for $n>N$.
What I am confused about is as to where he got $|s_n-s|<\sqrt{s}\epsilon$. If you reorganize the equation above, you get $\sqrt{s}|\sqrt{s_n}-\sqrt{s}|\leq|s_n-s|$. I don't understand how $|s_n-s|<\sqrt{s}\epsilon$ when $\sqrt{s}|\sqrt{s_n}-\sqrt{s}|\leq|s_n-s|$ and $|s_n-s|<\epsilon$.
Thanks for any clarification. There is a good chance I misunderstood something up to this point so any help is much appreciated.

What I am confused about is as to where he got $|_−|<\sqrt{s} \epsilon$.

First, what he should have said was that for any $\epsilon' > 0$, we can define $\epsilon = \sqrt{s}\epsilon'$. Now the assumption in your first highlighted box shows that there's some number $N$ such that $n >  N$ implies 
$$
|s_n - s | < \epsilon = \sqrt{s} \epsilon'
$$
Now since $\epsilon'$ was an arbitrary variable name, we can replace it with the name $\epsilon$, and conclude that for any $\epsilon$, there's a number $N$ such that $n > N$ implies 
$$
|s_n - s | < \sqrt{s}\epsilon.
$$
Why did the author choose to establish this apparently odd fact? Because it's the fact that'll make it possible to derive the conclusion the author wants, namely that for any $\epsilon$, there's an $N$ such that $n > N$ implies
$$
|\sqrt{s_n} - \sqrt{s} | < \epsilon.
$$
How'd the author know to do this? The author probably worked backwards to find it; it's how a lot of proofs like this are developed. 
