Subgroup containment relation in the group of integers equals divisibility relation on generators

Statement
Let $$\mathbb{Z}$$ denote the fact about::group of integers under addition. Suppose $$H,K$$ are subgroups of $$\mathbb{Z}$$. Suppose $$H = m \mathbb{Z}$$ and $$K = n \mathbb{Z}$$. Then, $$H \le K$$ if and only if $$n| m$$, i.e., $$m$$ is a multiple of $$n$$.

Note that, because every nontrivial subgroup of the group of integers is cyclic on its smallest element, the subgroups $$H$$ and $$K$$ can be written in the form $$m\mathbb{Z}, n\mathbb{Z}$$ respectively.

Corollaries

 * Given two subgroups $$m\mathbb{Z}$$ and $$n\mathbb{Z}$$, their intersection is the subgroup generated by an element $$l$$ with the property that $$m | l, n | l$$, and if $$m | k, n|k$$, then $$l|k$$. Such an integer $$l$$ is termed a least common multiple of $$m$$ and $$n$$ (if we allow only nonnegative integers, then it is unique).
 * Given two subgroups $$m\mathbb{Z}$$ and $$n\mathbb{Z}$$, their join is the subgroup generated by an element $$d$$ with the property that $$d | m, d | n$$, and if $$c | m, c|n$$, then $$c | d$$. Such an integer $$d$$ is termed a greatest common divisor of $$m$$ and $$n$$ (if we allow only nonnegative integers, then it is unique).
 * The greatest common divisor of $$m$$ and $$n$$ can be written as $$am + bn$$ for some integers $$a$$ and $$b$$. That is because it is in the subgroup generated by $$m\mathbb{Z}$$ and $$n\mathbb{Z}$$.