PMO 2017 Problem 1

Given $n \in \mathbb{N}$, let $\sigma (n)$ denote the sum of the divisors of $n$ and $\phi (n)$ denote the number of integers $n \geq m$ for which $\gcd(m,n) = 1$. Show that for all $n \in \mathbb{N}$,

$$\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}$$

and determine when equality holds.