The Principle of Mathematical Induction

Real Analysis · Axiom Academy

LESSON The Principle of Mathematical Induction Two checks — one true starting point and one self-propagating step — prove a statement for infinitely many natural numbers at once. 1. Two Parts Prove Infinitely Many Cases Let P(n) be a statement about a natural number n . To prove P(n) holds for every , induction asks for just two things: a base case that gets the chain started, and an inductive step that carries truth from each n to the next. Base case — verify the starting value Inductive step — truth at k forces truth at k+1 2. Running the Engine: the Sum We prove . Watch the inductive step as a picture: the blocks already stack to the area — that is the hypothesis. Adding one more column of height k+1 must land us exactly on the next closed form. The left side is just 1 ; the right side is . They agree, so P(1) is true. Inductive step. Assume the hypothesis P(k) — that — and add the next term (k+1) to both sides: Factor out (k+1) from the right-hand side and simplify: That last expression is precisely P(k+1) with k+1 in place of n . So — the step closes, and by the principle the identity holds for every . The two parts are not interchangeable add-ons — each does a job the other cannot. The inductive step is a row of valid implications: each says if this rung is true, the next one is too. But a chain of "if–then" links carries nothing until something at the bottom is actually true. The base case is that anchor.

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