A proof is a logical and structured argument to show that a mathematical statement (or conjecture) is always true.

Proof by deduction starts from known facts or definitions (or theorems), then uses logical steps to reach the desired conclusion.

Statements which are assumed to be true include:

• All even numbers can be written as 2n.
• All odd numbers can be written as 2n+1.

Proof by deduction involves:

1. Start with a statement, and assume that it is true.
2. Use this statement to show that another statement must be true.
3. Finish with a statement of proof.

Example 1

Prove that $n^2-n$ is an even number for all values of $n$.

We want to prove that $n^2 - n$ is even for all integers $n$.

Observe that $n^2 - n = n(n - 1)$. This is the product of two consecutive integers: $n$ and $n-1$. In any pair of consecutive integers, one of them is even. Therefore, their product is divisible by 2, i.e., it is an even number.

Thus, $n^2 - n$ is even for all integer values of $n$.