Proof by Exhaustion

Proof by exhaustion is a method of proof in which you verify a statement by checking every possible case. Once you have confirmed that the statement holds in every case, the proof is complete. This technique is only practical when the number of cases is finite and manageable.

In the AQA A-Level Mathematics specification (7357), proof by exhaustion is listed as one of the four types of proof that students must understand and be able to apply. It is a legitimate and powerful method when used appropriately.

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When to Use Proof by Exhaustion

Proof by exhaustion is appropriate when:

• The statement applies to a finite, small set of cases.
• It is not immediately clear how to construct a deductive proof.
• The problem explicitly restricts the domain to a small set (e.g., "for n = 1, 2, 3, 4, 5").

It is not appropriate when:

• The domain is infinite (e.g., "for all positive integers").
• The number of cases is impractically large.