Fixed Point Theorems

Intro to Proofs · Axiom Academy

Stir your coffee, crumple a map, run a feedback loop — something always ends up exactly where it started. That guarantee is a fixed-point theorem. Lay a map of the city flat on a table in that city. Exactly one printed dot sits directly over the real place it marks — the you-are-here point. Push the math and that dot can never fully disappear. Three moves show why, and how to find it. Where the curve meets the diagonal A continuous function that sends every point of the line [0,1] back into [0,1] must cross the diagonal y=x somewhere — and that crossing point is one the function leaves exactly where it is . Bend the curve, then trace it and watch the crossing appear. Brouwer promises a fixed point exists — but how do you find it? If f is a contraction (it always shrinks distances), just feed its own output back in over and over. From any starting point, the steps spiral into the one fixed point. That's the Banach fixed-point theorem. The catch in Banach's theorem is the word contraction : the map must shrink distances by some factor k<1 . Drag k and watch the cost — a tight grip (small k ) snaps onto the fixed point in a few steps, a loose one ( k near 1) crawls, and at it never locks on at all.

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