I just got the new (August-September 2010) of the American Math Monthly. In it is the following delightful article "On Cantor's First Uncountability Proof, Pick's Theorem, and the irrationality of the Golden Ratio" http://www.math.jhu.edu/~wright/Cantor_Pick_Phi.pdf The authors (Mike Krebs and Thomas Wright) first discuss Cantor's first proof (given two years before his famous diagonalization argument) of the uncountability of the reals, which is based on an algorithm which takes as input a countable sequence of reals (a_n) with the "intermediate value property" (every term in the sequence lies strictly between two others) and produces two other sequences whose elements come from the sequence (a_n), called (b_n) and (c_n) which are both monotone of opposite growth (i.e. if the first in increasing then the second is decreasing, etc.) and on complementary sides (i.e. if the first is increasing then every term of the second is bigger than every term of the first, etc.). Cantor then shows that the LUB of the increasing sequence can't be in the the original sequence. The authors apply this algorithm to the sequence of rationals in (0,1) ordered by denominator, and then within constant denominator by numerator. What they first find, experimentally, is that the elements of (b_n) and (c_n) are all ratios of consecutive Fibonacci numbers. They prove this result by induction and an appeal to Pick's theorem (which gives the number of lattice points in the interior of a lattice polygon in terms of the area and the boundary), and thus use Cantor's original argument to prove the irrationality of the golden ratio. Victor