Lemma (Feller volume 1, 175p): As $x \to \infty$ $$1- \Phi(x) \sim x^{-1} \phi(x);$$ more precisely, the double inequality $$ (x^{-1} - x^{-3}) \phi(x) < 1- \Phi(x) < x^{-1}\phi(x)$$ holds for every $x > 0$.
Proof: Obviously $$(1-3x^{-4}) \phi(x) < \phi(x) < (1+x^{-2})\phi(x).$$ The members are the negatives of the derivatives of those in the the second display in the lemma, and this follows by integration between $x$ and $\infty$.
$\sim$ indicates that the ratio between both sides converges to $1$ as $x$ tends to infinity. $\Phi$ and $\phi$ denotes normal cdf and pdf, respectively. In the proof, the first inequality holds since $1-3x^{-4} < 1 < 1+ x^{-2}$ and $\phi(x)>0$ for all $x$. When I integrate $\phi(x)$ between $x$ and $\infty$, I can get $1- \Phi(x)$. But, I am not sure how the integration of the other sides $(1-3x^{-4}) \phi(x)$ and $(1+x^{-2})\phi(x)$ results in $(x^{-1} - x^{-3}) \phi(x)$ and $x^{-1}\phi(x)$, respectively. I am also wondering how the inequality in the lemma implies the first display in the lemma.
I would appreciate if you give some help.
