  Consider the Mills ratio corresponding to the standard Gaussian law,
$f(x)=\big(1-\Phi(x)\big)/\phi(x), \, x\ge 0$, where $\phi$ is the density
function of this law and $\Phi$ its cumulative distribution function. We prove
that this function is completely monotone. In the proof we obtain a sequence of
rational functions that are sharp bounds for $f$; it turns out that these
rational functions are the convergents of the continued fraction defined by
$f$, and provide an approximation procedure that allows to prove interesting
properties where $f$ or its derivatives are involved. As an application we show
that $1/f$ is strictly convex.
