Be a non-empty set of primes $A $. Let us define $A^{\otimes}$ as the set of numbers smooth over $A$, that are the naturals having all their prime divisors in $A$ (where $1$ is arbitrarily considered as smooth over any set).

Using an elementary proof, I have established that the following sums
$$ \sum_{p \in A} \frac{1}{p} \quad \quad \mathsf{and} \quad \quad \sum_{n \in A^{\otimes}} \frac{1}{n} $$
are of the same nature, i.e. either they both converge or they both diverge; also, if these sums converge, then
$$ \sum_{p \in A} \frac{1}{p} \; < \; \log \left( \sum_{n \in A^{\otimes}} \frac{1}{n} \right) \; < \; 2 \sum_{p \in A} \frac{1}{p} $$
I have been told$-$quite tersely$-$by an expert in number theory that proving these results is *"a very simple exercise"*. Now, as I consider my proof as non-obvious, would anyone be kind enough

- either to provide some references, should these results be well-known,
- or to give some clues that these are indeed "simple" to prove, possibly using common non-elementary techniques?

PS: Note that the former proposition has several interesting corollaries, e.g. deriving from Brun's theorem that the sum of the reciprocals of numbers smooth over twin primes is convergent (this was incidentally my initial motivating case study).