The Impossible Staircase

Intro to Proofs · Axiom Academy

Try to build a staircase that descends forever through the positive integers — and discover why it always slams into a floor. That impossibility is a proof technique. M. C. Escher drew a staircase that climbs forever and somehow never gets higher. Mathematics has its own impossible staircase — one that tries to descend forever through the whole numbers. Watch what happens to it, and you'll meet one of the sharpest tools in a prover's kit. Each step must land on a strictly smaller positive integer than the one above it. Watch the staircase march down from 100 — and watch it hit the floor at 1 , where there is simply no positive integer left to step onto. However you choose the steps, the descent can't last forever — it is trapped above a floor it must eventually hit. Drive the descent yourself. The slider picks the next value — it must be a positive integer strictly below where you are. Each step drops you onto the number line; no matter how greedily or gently you go, the room beneath you keeps shrinking toward 1 . "Steps left" is the most you could possibly have remaining — and it can only fall. A counter of positive integers that only goes down must run out. The fraction that never bottoms out Run the same game on positive fractions . Drag the handle to halve the current fraction . The denominator doubles, the value keeps dropping — but it is still a positive fraction , every time. There is no floor here, and no smallest positive fraction.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.