Divisibility-closedness is strongly join-closed in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$H_i, i \in I$$ are all divisibility-closed subgroups of $$G$$. Then, the join of subgroups $$\langle H_i \rangle_{i \in I}$$ is also a divisibility-closed subgroup of $$G$$.

Related facts

 * Powering-invariance is strongly join-closed in nilpotent group
 * Divisibility-closedness is not finite-join-closed (the examples for this are solvable, but cannot be nilpotent)
 * Divisibility-closedness is not finite-intersection-closed (there is an abelian example)
 * Powering-invariance is strongly intersection-closed

Facts used

 * 1) uses::Divisible subset generates divisible subgroup in nilpotent group

Proof
The proof follows from Fact (1): simply take the set-theoretic union of the subgroups as the "divisible subset" for the appropriate set of primes and argue that the subgroup generated by it is also appropriately divisible.