Is this proof acceptable?

Question

Proposition:

If $g$ is any common factor of $m$ and $n$ where $g,m,n \in$ $\mathbb N$ then $g \mid lcm(m,n)$

Proof:

As $m \mid lcm(m,n)$ and $n \mid lcm(m,n)$ by transitivity of divisibility $g \mid lcm(m,n)$ $$\tag*{$\blacksquare$}$$

Thanks!

Answer 1

You should probably add that it follows from the definition of least common multiple that $m \mid \mathrm{lcm}(m,n)$.