Proof (MEX-P1)

3 dot points across 3 inquiry questions, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we prove inequalities rigorously, and how do foundational results such as the arithmetic-geometric mean inequality follow from elementary algebra?

Prove inequalities using algebraic manipulation, the fact that squares are non-negative, and standard results such as the arithmetic mean-geometric mean inequality

Proving inequalities. The consider LHS minus RHS method, the non-negative square, the AM-GM inequality with both algebraic and geometric proofs, the Cauchy-Schwarz inequality at Extension 2 level, building new inequalities from old, and rigorous forwards structure, with verified worked examples.

13 min read →

How do rigorous logical arguments such as proof by contradiction and counterexample establish or refute mathematical statements?

Use the language of proof, prove results by contradiction and contrapositive, and disprove statements by counterexample

Nature of proof. Logical language, implication and equivalence, if and only if, direct proof, proof by contradiction, the contrapositive, disproof by counterexample, and the common logical fallacies, with rigorous worked examples.

12 min read →

How does mathematical induction prove statements about all positive integers, including inequalities, divisibility, and results needing more than one base case?

Prove results involving sums, divisibility and inequalities for all integers using the principle of mathematical induction

Further mathematical induction. The base case, induction hypothesis and inductive step, applied to series, divisibility and inequalities, plus strong induction with multiple base cases, induction in less obvious settings, and the ways an inductive proof can quietly fail, with rigorous worked examples.

14 min read →