Mathematicians who are finitists.

[Jehanne]

As p goes to infinity?  Define your universe for p?

p does not go anywhere. It would not even appear in the formalized theorem.

The only variables in the formalized theorem are the numerator and denominator of ε. p = f(ε) where f is the primitive recursive function which sends n to the smallest p such that 2^p > n. We could, however, just take f to be the successor function, and apply it to the numerator of ε. The proof would be fine, since 2^{m+1} >= 2^{(m + 1) / n} > n.

Here's what Professor Kenneth Rosen of Columbia University has to say about the (weak) principle of mathematical induction in his book, Discrete Mathematics, 8th edition: