I am self studying analytic number theory from Tom M Apostol and I have a doubt in proof of a theorem. I am adding it's image.
I have doubt in 4 th line of proof where Apostol writes the convergence is also uniform for all t.
Is convergence uniform due to the fact that Both of terms in product in RHS of 3 rd line are absolutely convergent and hence the Terms on RHS becomes constant and hence independent of t. Am I right?
If I am wrong, can someone please give correct explanation for deducing uniform convergence for the product?