I want to prove that $\lim_{x\to\infty}\pi(x)/\frac x{\log x}=1$, where $\pi(x)$ is the number of primes less than $x$.
Let $\psi(x)=\sum_{p\leq x}\log p$. It is stated that I only need to show $\pi(x)-\frac{\psi(x)}{\log x}=O(\frac{x}{(\log x)^2})$ as $x\to \infty$.
I wrote down explicitly what big O notation means, but I have trouble seeing why this is so. Specifically, I'm not sure about the speed of increasing of $\psi(x)$ as $x\to \infty$. Could you give me any insight? Thanks!