An example in proving the limit of a complex function says:

Let us show that if in the open disk , then , the point being on the boundary of the domain of definition of .

Observe that when is in the disk , .

Hence, for any such and each positive , whenever .

We simply take .

I understand most parts of the proof. However, I don't know how they got to .

Isn't it that ?

I'm guessing it's because of the domain of definition of , but I don't know how this makes .