As usual define

\[\Phi(z) = \int_{-\infty}^z \phi(x) dx \quad\text{where} \quad \phi(x)=\frac{1}{2\pi} e^{-\frac12 x^2}\]

Some times it is use full to have an estimate of \(1-\Phi(z)\) which rigorously bounds it from above (since we can not write formulas for \(\Phi(z)\) ).

Follow the following steps to prove that

\[ 1- \Phi(z) < \frac{\phi(z)}{z}\,.\]

First argue that

\[ 1- \Phi(z) < \int^{\infty}_z \frac{x}{z}\phi(x) dx\,.\]

Then evaluate the integral on the right hand side to obtain the bound.