Direct proof of irrationality?

Below is a simple direct proof that I found as a teenager:

THEOREM $\;\rm r = \sqrt{n}\;$ is integral if rational, for $\;\rm n\in\mathbb{N}$.

Proof: $\;\rm r = a/b,\;\; {\text gcd}(a,b) = 1 \implies ad-bc = 1\;$ for some $\rm c,d \in \mathbb{Z}$, by Bezout

so: $\;\rm 0 = (a-br) (c+dr) = ac-bdn + r \implies r \in \mathbb{Z} \quad\square$

Nowadays my favorite proof is the 1-line gem using Dedekind's conductor ideal - which, as I explained at length elsewhere, beautifully encapsulates the descent in ad-hoc "elementary" irrationality proofs.

Wikipedia has a constructive proof. You can bound $\sqrt 2$ away from $p/q$.

Rational numbers have finite continuued fractions.

$\sqrt{2}=1+1/(\sqrt{2}+1)=1+1/(2+1/(\sqrt{2}+1))=\cdots$ Then the continued fraction is not finite 1+1/2+1/2+1/2+...

The geometric proof (not the one in Wikipedia), the one that proves $\sqrt{2}$ is not commensurable with $1$ is also direct (and is essentially the same as the continued fraction).