The n-th prime is less than $n^2$?

In Zagier's the first 50 million prime numbers a very elementary proof is given that for $n > 200$ we have

$$\pi(n) \ge \frac23 \frac{n}{\log n}$$

where $\pi(x)$ is the number of primes below $x$ (as well as a bound in the other direction). In fact, it already holds for $n \ge 3$, as can be directly checked.

Suppose that $p_n > n^2$, then for this $n$ we have $\pi(n^2) < n$, but that violates the bound already for $n = 5$.