State the prime number theorem, and Bertrand's postulate.

Let $S$ be a finite set of prime numbers, and write $f_{s}(x)$ for the number of positive integers no larger than $x$, all of whose prime factors belong to $S$. Prove that

$$f_{s}(x) \leqslant 2^{\#(S)} \sqrt{x}$$

where $\#(S)$ denotes the number of elements in $S$. Deduce that, if $x$ is a strictly positive integer, we have

$$\pi(x) \geqslant \frac{\log x}{2 \log 2} .$$