Q:

Question about distribution of primes

The following is from "Introduction to Number Theory" by Hardy and Wright.
The book first states the following theorem

Theorem A: If $\pi(x)$ is number of primes not exceeding $x$ then $\pi(x) \sim \frac{x}{\log{x}}$.

where $\sim$ stands for asymptotically similar. The book then states that the inverse of the function $\frac{x}{\log x}$ is asymptotic to the function $x \log {x}$ and thus Theorem A gives the following theorem

Theorem B: $p_n \sim n\log{n}$.

where $p_n$ is the $n^{th}$ prime number. I am unable to understand how we get Theorem B from Theorem A ? Am I missing something trivial ?  

A:

By definition of the prime counting function we have $\pi(p_n)=n$ for the $n$-th prime $p_n$. So, for $n\to \infty$,
$$
n=\pi(p_n)\sim \frac{p_n}{\log(p_n)},
$$
which says $p_n\sim n\log(n)$.

