Proof Techniques: Direct, Contrapositive, Contradiction

Most claims a quant needs to justify, "this estimator is unbiased," "this algorithm always terminates", take the form "if A, then B." There are three standard templates for proving that. A direct proof starts from A and reasons forward step by step until it reaches B. A proof by contrapositive proves the logically equivalent statement "if not B, then not A" instead, useful when it's easier to reason from B's negation than from A itself. A proof by contradiction assumes A is true but B is false, and derives a logical impossibility from that combination, which forces the conclusion that B must hold whenever A does.

All three prove exactly the same thing; the choice is about which direction is easier to argue, not about which is more rigorous.

A worked example: prove that if $n^2$ is even, then $n$ is even. A direct proof struggles here, knowing $n^2$ is even doesn't obviously hand you a clean expression for $n$. The contrapositive is easier: assume $n$ is odd, so $n = 2k+1$ for some integer $k$; then $n^2 = 4k^2 + 4k + 1$, which is odd. Since "$n$ odd implies $n^2$ odd" is proven directly, the original contrapositive statement "$n^2$ even implies $n$ even" is established.

A proof by contradiction of the same fact looks different: assume $n^2$ is even but $n$ is odd. Since $n$ is odd, $n^2$ is odd (shown above), contradicting the assumption that $n^2$ is even. The contradiction forces the conclusion that $n$ cannot be odd, so $n$ must be even. Notice this is really the contrapositive argument wearing a different logical wrapper, in practice the two often overlap, and picking between them is a matter of which framing makes the "impossible" step easiest to spot.