A universal theorem over a finite universe can sometimes be proved by checking every case. This method is called proof by exhaustion. It is useful only when the universe is explicitly finite and small enough for all cases to be verified. In this lesson, proof by exhaustion and theorem terminology are introduced through finite numerical examples and the language of theorems and corollaries.

A proof by exhaustion is a proof method used for a finite universe. Given a finite universe, to prove that a statement is true for every element of the universe, one verifies the statement separately for each element. If $$\begin{equation*} U=\{a_1,a_2,\ldots,a_n\}, \end{equation*}$$ then to prove $\forall x\in U,\ p(x)$, it is enough to show $$\begin{equation*} p(a_1),p(a_2),\ldots,p(a_n). \end{equation*}$$