In Churchill's book of Complex Analysis there are two statements that I can't match them to be consistent: In one place it says that a function must be analytic at a removable singular point :
why "must"? Because after removing singularity it becomes a series of positive powers.
But the following lemma say that if the function is not analytic at $z_0$ then definitely it has a removable singularity there:
and so must be analytic; but by the lemma it is not analytic! Where am I wrong?
Here is the proof of the lemma:
I don't understand why it supposes not being analytic then it arrvies at a Taylor series which implies analyticity?