[johnnny]

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.

[janm31415]

correct, I missed that.

[johnnny]

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.