Doubt in proof of theorem proving analyticity of $\Delta(\tau) $ in H.

Question

While studying Number theory from Apostol's Modular functions and Dirichlet series in number theory I have a doubt in proof of theorem 1.15 .

It's image is ->

I have doubt in line 11 when Apostol says that he will invoke lemma 1 . But the series in lemma 1 converges absolutely iff $\alpha$>2 . Then why he proves for $\alpha$ =2 ?

Adding image of Lemma 2 -

Can someone please explain.

Answer 1

Take the inequality after (6) to the power $-\alpha/2$, then you get (6) with $M=K^{-\alpha/2}$.