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 whereby we use algebraic symbols to represent any number and construct logical arguments from known facts to show that something is true for the symbol and this infers that it is true for all numbers. Proof by exhaustion is where we show that a statement is true for each number in consideration. This is also known as proof by cases – see Example 2. Proof by exhaustion also includes proof whereby numbers are split into a set of exhaustive categories and the statement is shown to be true for each category – see Example 1.

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.