In his 1999 review of Edward Tufte's Visual Explanations in the Notices of the AMS (third page), Bill Casselman gives a very pretty proof of the irrationality of the golden mean. More precisely, Casselman shows that the side and diagonal of a regular pentagon cannot be commensurable. The proof assumes the contrary, and thereby constructs a vanishing sequence of commensurable terms, which is a contradiction.
Upon closer inspection, however, I realized that there's a step in the proof that I cannot justify to my satisfaction. Casselman does show that the commensurability of the side and the diagonal of a regular pentagon implies the existence of a strictly decreasing sequence of commensurable lengths (e.g. the sequence consisting of the lengths of the sides of successively constructed nested regular pentagons). It is not clear to me, however, that he showed that this sequence converges to 0 (and not to some positive number).
Of course, convergence to 0 would follow immediately if we could establish that the sequence of side lengths is a geometric one, i.e. that successive terms in it are in a constant ratio to each other (which has to be $< 1$, since the sequence is strictly decreasing). That this ratio is constant seems to me intuitively obvious somehow, but cannot I prove it rigorously.
Any help would be appreciated!