(a) Show that $\int_2^x\frac{\pi(t)}{t^2}dt=\sum_{p\leq x }\frac{1}{p}+o(1)\sim\log\log x.$
(b) Let $\rho(x)$ be the ratio of the two functions involved in the prime number theorem:
$\rho(x)=\frac{\pi(x)}{x/ \log x}$ Show that for no $\delta>0$ is there a $T=T(\delta)$ such that $\rho(x)>1+\delta$ for all $x>T$, nor is there a $T$ such that $\rho(x)<1-\delta$ for all $x>T$. This means that $\lim \inf \rho(x)\leq 1 \leq \lim \sup \rho(x),$ so that if $\lim \rho(x)$ exists, it must have the value $1$.
I don't know how to apply (a) to (b), and I couldn't find any sources related to such a proof. Could you give me a proof on that?