Context: Introduction to SDEs, by Evans, page 43, proof of Lemma 3.
The author writes
$$\frac{2}{\sqrt{2\pi}}\int_x^{\infty}e^{-s^2/2}\mathrm{d}s\leq \frac{2}{\sqrt{2\pi}}e^{-x^2/4}\int_x^{\infty}e^{-s^2/4}\mathrm{d}s.$$
Why does this inequality hold? What does it follow from?
If $s\geq x$ then $$e^{-\frac{s^2}{2}}=e^{-\frac{s^2}{4}}e^{-\frac{s^2}{4}}\leq e^{-\frac{x^2}{4}}e^{-\frac{s^2}{4}}$$ and the result follows by multiplying both sides by $\frac{2}{\sqrt{2\pi}}$ and integrating.