Proof by cases partitions the hypothesis into exhaustive cases and proves the conclusion for each. If true in all cases, it's true in general. This method is essential when a universal approach is infeasible.
Identify natural partitions: parity (even/odd), sign (positive/negative/zero), or other exhaustive categories. Verify no cases are missed.
You've learned to write direct proofs, where you start from the hypothesis and derive the conclusion through a chain of logical steps. Proof by cases extends this when a single chain of reasoning can't handle all the ways the hypothesis can be true. The strategy is to partition all possibilities into a finite set of cases, prove the conclusion within each case separately, and then conclude that since the cases are exhaustive — every instance falls into at least one — the result holds in general.
The critical move is identifying the right partition. The partition must be exhaustive: every possible instance must fall into at least one of your cases; no situation should be left unaddressed. Common natural partitions are: parity (n is even vs. n is odd), sign (x > 0, x = 0, x < 0), or membership (n is in some set vs. n is not). You don't need the cases to be mutually exclusive — overlap is permitted — but you do need to handle each case completely.
Consider proving that n² + n is always even. You could factor: n² + n = n(n+1), and observe that consecutive integers always include one even number. But proof by cases makes this immediate. *Case 1: n is even.* Then n = 2k, so n(n+1) = 2k(2k+1) = 2·[k(2k+1)], which is even. *Case 2: n is odd.* Then n+1 is even, so n+1 = 2m, and n(n+1) = n·2m = 2·[nm], which is even. Since every integer is either even or odd, the cases are exhaustive, and the proof is complete. No algebraic ingenuity was required — just systematic enumeration.
An important mindset shift: proof by cases is not a fallback when you can't find a "better" proof. It is often the *cleanest* and most transparent proof available, especially in number theory and combinatorics. What distinguishes a rigorous case proof from hand-waving is the explicit verification that cases are exhaustive. A proof that leaves an edge case unhandled — say, forgetting n = 0 when proving something about non-negative integers — has a genuine gap, even if all other cases are airtight.