Revise Proof: induction, contradiction and counterexample for IB Mathematics: Analysis and Approaches (1.5)

A mathematical proof is a finite chain of logically valid steps that starts from things already accepted as true (definitions, axioms, earlier results) and ends at the statement you want to establish. If the steps are valid and the starting points are true, the conclusion cannot be false. This is a far higher standard than evidence. Checking that a formula works for $n=1,2,3,\dots,100$ shows only that it works for those hundred values; a universal claim ranges over infinitely many, so no finite check can settle it. That gap is exactly why the three techniques of this lesson exist: induction to cover all positive integers in one argument, contradiction to pin down properties that resist direct attack, and the counterexample to demolish a false universal claim with a single case.

This is the quickest technique and it does exactly one job: it disproves. If a statement claims something holds for every value in a set, a single case where it fails — a counterexample — kills the claim. Crucially, the reverse is not true: no number of successful cases can prove a universal statement, so a counterexample can never be used to prove, only to disprove.

To prove a statement by contradiction, you assume its negation — that it is false — and reason validly until you reach something logically impossible. Because your reasoning was sound, the only thing that can have gone wrong is the assumption, so the original statement must be true. This technique shines where a direct attack is awkward: proving a number is irrational, or that infinitely many objects exist. The set-piece example every HL student should know cold is the irrationality of $\sqrt{2}$.

Induction proves that a proposition $P(n)$ holds for every positive integer $n$ from a starting value onward. The idea is the falling dominoes: if the first domino falls (the base case), and every fallen domino topples the next (the inductive step), then all of them fall. Formally there are four parts, and each carries marks: define $P(n)$, prove the base case, state the inductive hypothesis for $n=k$, complete the inductive step to $n=k+1$, and write the concluding statement. The syllabus applies this to three flavours of statement — summation formulae, divisibility results and inequalities — and the structure is identical in all three.

A proof by induction has a standard mark allocation, and our engine mirrors it exactly. Because the result is 'answer given' (AG), the marks live in the structure: method marks (M) for each required move, an accuracy mark (A) for the key algebra, and reasoning marks (R) for the valid inductive step and the concluding statement. Study how each of the six marks below attaches to a specific line — that is precisely how the Practice button will mark your own attempt.

The proof habits you build here recur across HL: induction reappears for de Moivre's theorem in complex numbers, for results about derivatives and sequences, and for many series identities; contradiction underlies uniqueness and irrationality arguments; and the counterexample remains your fastest tool for testing whether a tempting conjecture is actually true. More than any single result, what transfers is the discipline of the four techniques — state the proposition, cover the base case, use the hypothesis, name the impossibility, exhibit the witness, and always close the loop. Master that structure and proof questions become variations on a method you already own.