I'm very new to number theory and looking for a proof of the following inequality:
$$c' \log^{\text{#} \mathbb{P}}{R} \leq \sum \limits_{\substack{n \leq R\\p|n \implies p \in \Bbb P}} 1 \leq c \log^{\text{#}\Bbb P}{R}$$
where $R$ is a positive integer, $p$ is a prime number, $\Bbb P$ is a finite set of primes, $\text{#} \Bbb P$ is the number of elements in $\Bbb P$, and $c$ and $c'$ are independent constants.
Any help would be great. Any opinion where should I start? Thanks.