3 ms·

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

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 4y ago
correct, I missed that.
- johnnny 4y ago
  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.