Divisibility-closedness is transitive

Statement
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a divisibility-closed subgroup of $$K$$ and $$K$$ is a divisibility-closed subgroup of $$G$$. Then, $$H$$ is a divisibility-closed subgroup of $$G$$.

Related facts

 * Powering-invariance is transitive