https://www.jamp-audio.com I started making audio plugins for iOS 2 years ago. I'm making about $300/month.

I might be missing something, but as I see it here: by assumption, √2 is rational. Then k(√2-1) √2 is a positive integer.

This does not mean that k(√2 - 1) belongs to K, with K the set of positive integers {n: n√2 ∈ ℤ}, as k(√2 - 1) is not necessarily an integer.

The proof can be fixed I think with K the set of rational numbers {q: q√2 ∈ ℤ}

k * √2 is an integer, and k is an integer, by construction of K and the initial assumption that K is not empty.

Therefore their difference k * √2 - k = k * (√2 - 1) is an integer.

correct, I missed that.

As fro your further edit: assuming √2 is rational, then the set of ~~rational~~ integers {q: q√2 ∈ ℤ} does not admit a minimum, indeed it contains arbitrarily small numbers, so we can't pick k the smallest member of that set.