Answer
Proof below.
Work Step by Step
This is a direct proof.
Let $x$ be a rational number expressed as the fraction $\frac{a}{b}$, with $a, b$ integers. Now, consider $1/x$. This is defined, because $x\neq 0$. $1/x=\frac{b}{a}$. Because $a, b\in \mathbb{Z}$, $1/x$ is rational.
