Mark Dominus on a proof by Tom M. Apostol published in 2000:
In short, if √2 were rational, we could construct an isosceles right
triangle with integer sides. Given one such triangle, it is possible
to construct another that is smaller. Repeating the construction, we
could construct arbitrarily small integer triangles. But this is
impossible since there is a lower limit on how small a triangle can be
and still have integer sides.