Proof by Exhaustion
When a claim only has to hold for a finite, manageable set of cases, you can prove it by simply checking every single one — leaving no case unexamined.
Most proofs have to cover infinitely many cases at once — that's why we use a general $n$. But some claims only need to be true for a finite number of possibilities. When that's the case, there's a blunt but perfectly valid strategy: check every possibility, one by one. If the claim holds in every single case, and you've genuinely covered them all, then it's proven. This is proof by exhaustion — you exhaust the possibilities.
The name is literal. To "exhaust" a set of cases means to use them all up, leaving none behind. The entire validity of the method rests on one condition: you must be certain you have covered every case. Miss one, and the proof collapses — because that one unchecked case might be the exception.
It feels almost too simple — "I just checked them all?" But think about what proof means: showing a statement is true in every situation it claims to cover. If a claim only covers, say, the integers from 1 to 5, then checking those five cases isn't a sample — it is the whole truth. There are no other cases hiding. That's the crucial difference from testing examples in an infinite claim, where you can never check them all.
When you're allowed to use it
Exhaustion only works when the number of cases is finite and small enough to list. You can use it when:
• the claim is about a specific, limited range (e.g. "for all integers $n$ with $1 \leq n \leq 5$"), or
• the possibilities naturally split into a few distinct types that together cover everything (e.g. "every integer is either even or odd" — just two cases to check).
That second kind is the clever one. Even when there are infinitely many numbers, they might fall into just two or three categories, and you can check each category. We'll see that in example 3.
Exhaustion is only as good as your case list. If you claim to have covered everything but secretly missed a case, the "proof" proves nothing. Before writing the final line, always ask: "Have I genuinely accounted for every possibility?"
Worked example 1
The final two lines matter as much as the checking. Step 4 explicitly states that every case is covered — that's what turns a list of calculations into a proof. Never just show the arithmetic and stop.
Worked example 2
Notice this only proves it for $n = 1$ to $4$. The result is actually true for all integers (because $n^2+n = n(n+1)$ is a product of consecutive integers, one of which must be even) — but exhaustion can't show that, since there are infinitely many integers. This is a perfect illustration of exhaustion's limit: it's watertight over a finite range, and useless beyond it.
Worked example 3
Here there are infinitely many integers — so we can't check them one by one. But every integer is either even or odd. Those two categories cover all integers, so if the claim holds for both, it holds for everything. Two cases to exhaust.
This is exhaustion at its most powerful. We couldn't check infinitely many integers — but we could check the two types every integer belongs to. The whole method hinges on step 3: proving the cases genuinely cover everything. Choosing the right way to split into cases is the real skill.
Check yourself
Prove that every integer $n$ with $1 \leq n \leq 5$ satisfies $n^3 \geq n^2$.