By way of contradiction, say there are positive integers without prime factorizations. Then there is a smallest such integer; call it N. N is not prime, because then it would have a prime factorization. So N has some divisor a such that 1 < a < N, and we can write N = ab for some integers a, b greater than 1. By assumption, N was the smallest integer without a prime factorization, so a and b have prime factorizations and we can concatenate these to get a prime factorization of N.I want to bring your attention to the bolded "we can write", which I definitely said while presenting the proof. (The other language might not be exactly what I used.)
Sure, we can write that. But we can also write "2 + 2 = 5". Or we could have just written "N = ab" at the beginning When a mathematician says "we can write X" for some statement X, they mean something like "X is true, for suitable values of some variables which might be contained in X that we haven't mentioned yet, and which we'll talk about now."
In short, mathematicians are only capable of writing true things, or so we'd want people to think from our writing. If only it were so easy!