Proof by Exhaustion

In maths, proof by exhaustion is proving 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 – see Example 1.
Proof by Exhaustion is different from proof by deduction whereby we use algebraic symbols and construct logical arguments from known facts to show that something is true for all numbers. Proof by exhaustion is where we show that a statement is true for each number in consideration.
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.


Example 1

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




Example 2

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.