**Proof by Exhaustion** is the proof that something is true by showing that it is true for each and every case that could possibly be considered. This is also known as **Proof by Cases**. This is different from Proof by Deduction where we use algebraic symbols and construct logical arguments from known facts to show that something is true for all numbers. For the case of Proof by Exhaustion, we show that a statement is true for each number in consideration (or subsets of numbers) – see Example 1 in the AS Maths Proof course below.

Proof by Exhaustion also includes proof where numbers are split into a set of exhaustive categories and the statement is shown to be true for each category. Proof by Deduction can then be used within the categories – see Example 2 in the AS Maths Proof course below.