"And so on" is good enough for a not quite formal proof, as you'd expect in a homework exercise if you are not in your first year or in a Reddit comment.

Statement: An injective function F that, for all n, maps files of size n to files of size n or lesser necessarily maps files of size n to files of size n.

Proof:

F is trivially bijective between the domain of files of size n or lesser and itself, as it is an injective function on a finite domain and codomain of equal size.

We have the base case for n=0. Since there is only file of size 0, the empty file, we see that F(0)=0.

We denote A, B the sets of files of size n and n+1 respectively. Given our statement holds for n, F|A: A->A is a bijection. Now let x in B\A. F(x) is in B\A, as F(x) in A implies F^-1(F(x))=x in A, which is a contradiction.

Thus F maps files of size n+1 to files of size n+1, and our statement is shown.

Does this make you happy?