What a Proof Is

A proof isn't intimidating ceremony — it's an argument so airtight it leaves no room for doubt: a chain of valid steps from things you already accept to the thing you're claiming. Here's how proofs actually work, including the famous techniques.

1. What a Proof Actually Is A proof is a gap-free chain of valid steps from things you already accept (axioms, definitions, known facts) to the claim you want to establish. Unlike evidence, a correct proof makes the conclusion certain.
2. The Main Proof Techniques Most proofs use one of a few shapes: prove it directly, assume the opposite and hit a contradiction, prove the contrapositive instead, split into cases, or — to disprove a universal claim — produce a single counterexample.
3. Proof by Induction Induction proves a statement for all natural numbers with two steps: show it holds for the first case, then show that if it holds for one case it holds for the next. Like dominoes — knock the first, and the rule topples the rest.