Prove that

Let $\frac{1}{\sqrt{3}}$ be rational.

$\therefore \frac{1}{\sqrt{3}}=\frac{a}{b}$, where $a, b$ are positive integers having no common factor other than 1

$\therefore \sqrt{3}=\frac{b}{a}$         .................(1)

Since $a, b$ are non-zero integers, $\frac{b}{a}$ is rational.

Thus, equation (1) shows that $\sqrt{3}$ is rational.

This contradicts the fact that $\sqrt{3}$ is rational.

Hence, $\frac{1}{\sqrt{3}}$ is irrational.