This question already has an answer here:
Weinberg uses the LSZ reduction formula to introduce field renormalization,and on page 441, he says:
As this discussion should make clear: the renormalization of masses and fields has nothing to do with the presence of infinities, and would be necessary even in a theory in which all momentum space integrals were convergent.
I believe weinberg is right , but I can't convince other people: how is renormaliztion needed without infinities? What does it have to do with the LSZ formula? I want answers to make clear the field renormalization with LSZ reduction formula. so maybe the question is different , because I am asking weinberg's book.