Direct Proof

A direct proof starts from the hypotheses and derives the conclusion through definitions, algebra, and previously established results. Its basic pattern is:

1. State the hypotheses precisely.
2. Apply definitions and valid inference rules.
3. Reach the desired conclusion.

ExampleThe square of an odd integer is odd

Assume that $n$ is odd. Then $n=2k+1$ for some integer $k$. Therefore

$$n^2=(2k+1)^2=4k^2+4k+1=2(2k^2+2k)+1.$$

Because $2k^2+2k$ is an integer, this has the form $2m+1$. Hence $n^2$ is odd.

The proof works because it uses the definition of oddness rather than relying on a numerical pattern.

ExerciseThe sum of two even integers

Prove that the sum of two even integers is even.

Solution

Let $a=2k$ and $b=2m$, where $k,m\in\mathbb{Z}$. Then

$$a+b=2k+2m=2(k+m),$$

and $k+m$ is an integer. Thus $a+b$ is even.

ExerciseA divisibility example

Prove that $n^3+2n$ is divisible by $3$ for every positive integer $n$.

Solution

Rewrite the expression as

$$n^3+2n=n(n^2-1)+3n=n(n-1)(n+1)+3n.$$

Among three consecutive integers, one is divisible by $3$. Hence both terms on the right are divisible by $3$, so the original expression is divisible by $3$.

The main difficulty in a direct proof is usually choosing a representation that exposes the relevant definition or divisibility property.