Let $x$ be a real number greater than or equal to 2 , and define

$$P(x)=\prod_{p \leqslant x}\left(1-\frac{1}{p}\right)$$

where the product is taken over all primes $p$ which are less than or equal to $x$. Prove that $P(x) \rightarrow 0$ as $x \rightarrow \infty$, and deduce that $\sum_{p} \frac{1}{p}$ diverges when the summation is taken over all primes $p$.