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Proof by cases

The idea in proof by cases is to break a proof down into two or more cases and to prove that the claim holds in every case. In each case, you add the condition associated with that case to the fact bank for that case only. As long as the cases cover every possibility, you have proved the claim regardless of what the actual case is. Proof by cases is closely related to the idea of using If-statements in a computer program.


Theorem. For every integer n, n2n.

Proof.. By cases. There are three cases: n ≤ −1, n = 0 and n ≥ 1.

Case 1. (n == 0). Notice that 02 ≥ 0, so the theorem holds when n = 0.

Case 2. (n ≥ 1).

(1) n ≥ 1 (from the condition for this case)
(2) n2n (multiply both sides of (1) by the positive value n)
So the theorem holds in this case.

Case 3. (n ≤ −1). Since n ≤ −1 and n2 ≥ 0, the theorem clearly holds in this case.