Let $x > 0$. By writing that $\varphi ( t ) \leqslant \frac { t } { x } \varphi ( t )$ for all $t \geqslant x$, show that
$$\int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t \leqslant \frac { \varphi ( x ) } { x }$$
where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.