real analysis - square root of 2 irrational - alternative proof

I have found the following alternative proof online.

It looks amazingly elegant but I wonder if it is correct.

I mean: should it not state that $(\sqrt{2}-1)\cdot k \in \mathbb{N}$ to be able to talk about a contradiction?

Doesn anybody know who thought of this proof (who should I credit)? I couldn't find a reference on the web.

Answer

The proof is correct, but you could say it's skipping over a couple of steps: In addition to pointing out that $(\sqrt2-1)k\in\mathbb{N}$ (because $\sqrt2k\in\mathbb{N}$ and $k\in\mathbb{N}$), one might also want to note that $1\lt\sqrt2\lt2$, so that $0\lt\sqrt2-1\lt1$, which gives the contradiction $0\lt(\sqrt2-1)k\lt k$.

As for the source of the proof, you might try looking at the references in an article on the square root of 2 by Martin Gardner, which appeared in Math Horizons in 1997.