[LibWhacker]

I don't see how it follows that there are some that require an infinite proof.

The reasoning is similar to the proof that the number of primes is infinite. You start by assuming that there is some largest prime and show that that leads to a contradiction. We can do the same thing here . . . Assume that there is some "largest proof," and you'll see this leads to an immediate contradiction.

[Virginia-American]

The number of theorems that can be proved is infinite, but each proof is finite.

If assuming that there is a longest proof leads to a contradiction, then there is no longest one. That doesn't imply that any of them is infinite; by analogy, no prime number is infinite.

[Doctor Stochastic]

Note that in your comments, you have proved that there are an infinite number of primes, not that there is an infinitely large prime.

(It's possible that) one may show that for any N, there is a proof that requires N+1 steps, but that doesn't mean that there is a proof that requires infinitely many steps. To show that, one must exhibit a statement that, for any N, cannot be proved in N or fewer steps.