I am self studying analytic number theory from Tom M Apostol Modular functions and Dirichlet series in number theory and I couldn't think about an argument in proof related to Modular group.
I am adding image of proof highlighting the argument which I don't understand.
My doubt is in 7 th line of 2nd image ie
$\phi $ is bounded in $R_\Gamma$ and it has been proved that $\phi $ is invariant under $\Gamma$ , so how does Apostol deduces $\phi $ is bounded in H?
Can someone please give a hint.