Simple Proof Methods

Before jumping into a few more involved proof methods, let's first see a couple situations where the proof is easier than we've imagined.

A trivial proof is a proof in which the consequent is always true, regardless of the antecedent. This occurs when the statement to be proven holds in all cases.

Example: Prove that for any even integer $n$, we have $n^2 + 1 > 0$.

Proof. The consequent $n^2 + 1 > 0$ is always true for any integer $n$ because squaring any integer results in a non-negative number, and then adding $1$ ensures the result is always positive. Since the conclusion is true regardless of whether $n$ is even, the statement is trivially true (so the proof is called trivial.)◼