## Proof by Exhaustion

In maths, proof by exhaustion means that you can prove that something is true by showing that it is true for each and every case that could possibly be considered. This is different to proof by deduction in that we do not use algebraic symbols to represent any number and showing that it is true for the symbol infers that it is true for all numbers – we must show that it is true for each number in consideration.

### Example 1

Prove that every perfect cube number is a multiple of 9, one less than a multiple of 9 or one more than a multiple of 9.

### Example 2

Prove that $(n+1)^3\geq 3^n$ for $n\in{\mathbb N}, n\leq 4$.