Mathematicians who are finitists.

[Jehanne]

Here's a proof by John Hopcroft, Jeffrey Ullman and Rajeev Motwani in their Introduction to Automata Theory, Languages, and Computation Edition 3 textbook, page 9:

Sorry, but it doesn't prove what you claimed. This proposition says that if S is finite and U is infinite, then T=U-S,
the complement of S in U, is infinite.

This does NOT prove the existence of an infinite set. It shows that *if* there is an inifnite set U, then there are other
infinite sets (U-S where S is finite).

Now, show how to get that set U from finitist principles.

Proofs that there are infinitely many primes