See https://math.stackexchange.com/questions/4216831/what-does-c... for some discussion on convergence using constructive mathematics.

It's the definition of "exists" that is contested. To me, the infinity of anything is a generator (a program that continues to print digits). Does some infinite number exists "now"? Not really (it's being printed). But is the number being printed larger than any given integer? Yes. So, does infinity exists? ¯\_(ツ)_/¯

"The Emperor's New Mind" is a great book on this and related topics.