Consider proving that for any integer n, the expression n² + n is always even. You could try algebraic manipulation, but there's a more elegant approach: what if n is even, and what if n is odd? By examining these two scenarios separately, the proof becomes almost trivial in each case.

Proof by cases embodies a fundamental principle of problem-solving: when a single argument seems impossible, partition the problem into manageable pieces. The technique transforms overwhelming complexity into a collection of simpler sub-problems, each yielding to focused reasoning.

Yet this apparent simplicity conceals critical requirements. Your cases must cover every possibility—leave a gap, and your proof crumbles. Choose your divisions poorly, and you've merely traded one hard problem for several hard problems. Master this technique, and you'll possess one of mathematics' most versatile tools for establishing certainty.

A proof by cases fails completely if the cases don't cover all possibilities. This isn't a minor flaw—it's a logical catastrophe. If your theorem claims something holds for all integers but your cases only address positive integers, you've proven nothing about the full claim.

The exhaustion requirement demands that every element in your domain falls into at least one case. Consider proving a property for real numbers. Dividing into 'positive' and 'negative' cases leaves a gap: zero belongs to neither. Your proof must either add a third case or redefine divisions as 'non-negative and negative' or 'positive and non-positive.'

Verification techniques vary by context. For finite domains, explicit enumeration works: proving something about days of the week requires exactly seven cases. For infinite domains, rely on logical dichotomies. Every integer is either even or odd—this exhausts all possibilities by definition. Every real number satisfies exactly one of x < 0, x = 0, or x > 0 by the trichotomy property.

When constructing case divisions, state explicitly why exhaustion holds. Phrases like 'since every integer is either even or odd' or 'by trichotomy of real numbers' signal to readers—and to yourself—that you've verified completeness. This habit prevents the embarrassing discovery that your cases missed something.

Takeaway

Before writing any sub-proof, verify that your cases cover every possibility in the domain—a proof with gaps isn't a proof at all.

Not all case divisions are equally useful. The goal isn't merely to partition your domain—it's to partition it in a way that makes each sub-proof tractable. Poor case selection transforms proof by cases from a problem-solving tool into busywork.

Consider proving that every prime greater than 3 leaves remainder 1 or 5 when divided by 6. You could examine six cases based on remainders 0 through 5, but this creates unnecessary work. Instead, observe that primes greater than 3 must be coprime to 6, immediately eliminating remainders 0, 2, 3, and 4. Two cases remain, and each yields quickly to direct verification.

The ideal case division satisfies two criteria: cases should be exhaustive (required) and cases should isolate the difficulty (strategic). When proving properties of functions, dividing by the sign of the input often helps because absolute value behavior changes at zero. When working with divisibility, dividing by remainders modulo relevant numbers exploits the algebraic structure.

Beware the trap of too many cases. A proof examining twenty separate scenarios might be correct but unilluminating—and tedious to verify. If you find yourself drowning in cases, step back. Is there a coarser division that still simplifies each sub-proof? Sometimes a single clever observation eliminates half your work.

Takeaway

Choose case divisions that exploit the problem's structure—the right partition makes each sub-proof straightforward, while the wrong one multiplies your difficulty.

Symmetry is the mathematician's shortcut. When two cases would require identical reasoning with variables swapped, you needn't write both proofs. The phrase 'without loss of generality' (often abbreviated WLOG) signals that you're leveraging symmetry to reduce cases.

Suppose you're proving a statement about two arbitrary real numbers a and b. If the statement is symmetric in a and b—meaning swapping them doesn't change anything relevant—then assuming a ≤ b loses no generality. Whatever holds for (a, b) with a ≤ b also holds for (b, a) by the same reasoning with labels exchanged.

However, WLOG arguments require genuine symmetry. Claiming 'without loss of generality, let n be even' when proving something about all integers is incorrect—evenness and oddness aren't interchangeable properties. The test: could you mechanically swap the cases and get valid reasoning? If not, you must prove each case separately.

Symmetry detection comes with practice. Look for interchangeable variables, symmetric functions, or geometric symmetry. In graph theory, recognizing that two vertices play identical structural roles halves your work. In analysis, even and odd function decomposition often reveals hidden symmetry. Each recognized symmetry is a case eliminated—proof by cases becomes proof by fewer cases.

Takeaway

Legitimate 'without loss of generality' arguments require true symmetry—the cases must be genuinely interchangeable through relabeling, not merely similar in difficulty.

Proof by cases succeeds through disciplined partition: exhaustive coverage ensures no possibilities escape, strategic selection makes sub-proofs tractable, and symmetry recognition eliminates redundant work.

The technique reflects a broader truth about mathematical reasoning. Certainty emerges not from handling everything at once, but from systematically handling everything—piece by verified piece, case by proven case.

As you encounter resistant theorems, ask yourself: what natural divisions exist? What cases would simplify the argument? The answer often transforms an impenetrable wall into a series of manageable steps.