Proof by Exhaustion

A proof by exhaustion splits a problem into a finite number of separate cases and then checks every single one. If the statement holds in all of them, it is proved. The catch is in the name: you can only do this when the cases are finite — you cannot exhaust infinitely many.

For example, suppose we claim that every integer from 1 to 5, when squared, is less than 30. There are only five cases, so we just check them all:

1^2 = 1,\quad 2^2 = 4,\quad 3^2 = 9,\quad 4^2 = 16,\quad 5^2 = 25

Every result is below 30, and there are no other cases to try, so the statement is proved. ✓

To prove a statement by exhaustion:

break the problem into a finite set of cases that together cover every possibility;
verify each case on its own;
if all cases pass, the statement is proved;
if even one case fails, the statement is false;
it is only usable when the cases are finite — you can't exhaust infinitely many.