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22 23 the InFInIte staIrcase a proof by regrouping A classical paradox involves a number of identical bricks that are stacked up on top of a desk, as in the diagrams opposite. It is easy to prove that by adding more and more bricks as indicated, we can make the resulting staircase protrude as much as we want. A staircase of n bricks, each of length 2, protrudes a distance of So, what we want to demonstrate is that the above sum approaches infinity as n does. Proof We first group the infinite sum as follows We replace every term by a number less than or equal to it to produce a new sum which is less than or equal to the one we started with, and we notice that our substitutes add up to infinity This means that the sum we are chasing is also infinite. Q.E.D. Note that the staircase also gets infinitely tall as it grows infinitely broad and that actually building it gets very tricky very fast, involving closer and closer spacings of the bricks. 23
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