Find the maximum of

${\displaystyle \frac{1}{p}+\frac{1}{q}+\frac{1}{r}}$

where $p$, $q$ and $r$ are positive integers and

${\displaystyle \frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1.}$

Prove that it is indeed a maximum.