Approximate Value of Nth Prime Number

Theorem
The $n$th prime number is approximately $n \ln n$.

Proof
This will be demonstrated by showing that:
 * $\displaystyle \lim_{n \mathop \to \infty} \dfrac {p_n} {n \ln n} = 1$

where $p_n$ denotes the $n$th prime number.

By definition of prime-counting function:
 * $\map \pi {p_n} = n$

The Prime Number Theorem gives:
 * $\displaystyle \lim_{x \mathop \to \infty} \dfrac {\map \pi x} {x / \ln x} = 1$

Thus:
 * $\displaystyle \lim_{x \mathop \to \infty} \dfrac n {p_n / \ln p_n} = 1$