I would like to prove the following bound:
$$\int_x^\infty e^{-y^2/2}\,dy\le \frac1x e^{-x^2/2}$$
Is it valid to say that $$x^{-1}\exp(-x^2/2) = \int_{x}^{\infty} \left(\frac{y^2+1}{y^2}\right)\exp(-y^2/2)\,dx \geq \int_{x}^{\infty}\exp(-y^2/2)\,dx\,?$$
If so, why do most solutions online and in textbooks (like this) avoid using this method? I have a feeling i'm doing something wrong.
Thanks!
