Why epsilon < 1 in this proof? In this proof, why is this stuff with $\delta _{1}$ and $\varepsilon = 1$ needed? I don't see how it is  necessary. Thank you.



 A: I'm not sure what you mean by "necessary". They are properly applying the techniques of proof, and the proof works.
But perhaps it is worthwhile examining the technique of proof that is being applied.
In that portion of the proof, the hypothesis is that (1) is true:

For every $\epsilon > 0$ there exists $\delta > 0$ such that if $|z-z_0|<\delta$ then $\left| \frac{f(z)-f(z_0)}{z-z_0} - L \right| < \epsilon$. 

So now one might ask: How does one apply this true statement?  
The answer is simple: You may apply it using whatever value of $\epsilon$ you like. 
For example, you may apply it with $\epsilon = .0000001$, and you will get a number $\delta$.
Or, you may apply it with $\epsilon = 1000000$, and you will get some other value of $\delta$ (depending on $1000000$).
The author chose to apply it with $\epsilon = 1$, and they got some value of $\delta$, which they chose to denote $\delta_1$. They then used that value of $\delta_1$, quickly renamed $r=\delta_1$, in the remainder of the proof.
In general, given a universally quantified statement "$\forall x \in A, P(x)$" which is known to be true, the method for applying it is that you may take any element $x \in A$. You then conclude that $P(x)$ is true. You may then subsequently apply $P(x)$ in your proof. Sometimes the subsequent proof depends on a clever choice of $x \in A$, other times the proof works no matter what element $x \in A$ you pick.
It is possible that for this theorem, the subsequent proof works no matter what value of $\epsilon > 0$ you pick, e.g. $\epsilon = .0000001$ or $\epsilon = 1000000$. Perhaps you can verify for yourself whether that is true. But even if that is true, it doesn't make the proof any less valid as written, it just means that there are uncountably many different valid proofs, one for each $\epsilon > 0$. 
I'm just glad that I don't have to rewrite uncountably many different proofs for each theorem. One proof is enough.
