$\therefore$Proof Writing & Problem Solving

Direct Proof

Start from what you know, walk forward one justified step at a time, and land exactly on what you want. Prove facts about odd and even numbers, split into cases, and make divisibility arguments airtight.

Contrapositive and Contradiction

When the front door is locked, try the back. Prove a statement by proving its contrapositive, or assume it is false and watch the world collapse. Show that √2 is irrational and that the primes never run out.

Induction

Knock over the first domino, show each one knocks over the next, and they all fall. Prove formulas for every n, use strong induction when one step back is not enough, and catch the famous fake proof that all horses are the same colour.

Pigeonhole and Extremal Arguments

Two of the cleverest tricks in proof writing. Too many pigeons force a crowded hole, and choosing the biggest, smallest or first of something often hands you the proof. Plus infinite descent: there is no smallest counterexample.

Existence, Uniqueness and Counterexamples

"For all" and "there exists" in the right order, one counterexample to sink a claim, how to prove there is exactly one, and a beautiful proof that something exists without ever finding it.

Problem-Solving Strategies

Pólya's four steps for getting unstuck, then three power tools: invariants (what never changes), colourings (what a chessboard pattern reveals), and monovariants (what only goes one way). Finish by beating a computer at Nim.

Writing It Up

A correct idea is only half a proof: the reader has to be able to follow it. Learn the style rules good mathematicians use, hunt the bug in a wrong proof, and structure a longer argument with lemmas.