Rational Numbers and Factorials.

First we can show that all prime numbers can be written as a quotient of products of factorials by induction. The first prime number, $2$ , is equal to $2!$ Then assuming this property is true for all prime numbers $p_n$ , $p_{n + 1}$ is also true, because $p_{n + 1} = \frac{p_{n + 1}!}{(p_{n + 1} - 1)!}$ , and $(p_{n + 1} - 1)!$ can be factored by prime numbers smaller than $p_{n + 1}$ , and those prime numbers (by assumption) can be written as a quotient of products of factorials.

Since every rational number can be written as a quotient of products of prime numbers, and every prime number can be written as a quotient of products of factorials, it follows that every rational number can be written as a quotient of products of factorials, which makes the statement true .