The Domino Effect

Real Analysis · Axiom Academy

A falling row of dominoes is the whole idea behind mathematical induction — one of the proof methods this course is built on. Topple the first, make sure each knocks the next, and infinitely many statements are proven at once. A claim like P(n) — "the statement is true for the natural number n " — has to hold for every : for 1 , for 2 , for 3 , on forever. You can never check infinitely many cases one at a time. So instead of checking, you build a chain. Line up the statements like dominoes: prove the first one true, and prove that whenever one is true the next must be too. Then you never have to look at each case — the whole infinite row is guaranteed to fall. Watch the perfect chain. The dominoes are evenly spaced, so each falling tile reaches the next. Press play to topple the first one and let the cascade run all the way to the end. The first falls (the base case) and each tile reaches the next (the inductive step), so the cascade never stops — that is exactly a proof by induction. What if one domino can't reach the next? Now widen the gap between one domino and the next, then push the first tile. If a falling domino can't reach its neighbor, the cascade stops dead right there. This is the heart of an induction proof: the inductive step , "if P(k) is true, then P(k+1) is true." Break that single link — fail to prove the implication for even one k — and every statement past it is left unproven.

This is the written version of the interactive lesson above. See the full Real Analysis course.