Let $\pi(x)$ be the number of primes $p \leqslant x$. State the Legendre formula, and prove that

$$\lim _{x \rightarrow \infty} \frac{\pi(x)}{x}=0 .$$

[You may use the formula

$$\prod_{p \leqslant x}(1-1 / p)^{-1} \geqslant \log x$$

without proof.]