Comment by tromp

The new proof rephrased:

Let K be the set of positive integers k such that k √2 is an integer.

Suppose K is non-empty, denote its minimum as k.

Consider the positive integer k' = k √2 - k = k (√2 - 1).

Then k' < k and k' is also in K since k' √2 = 2 k - k √2, contradicting non-emptyness of K.