positive multiple of an abundant number is abundant

Theorem. A positive multiple of an abundant number is abundant.
Proof. Let $n$ be abundant and $m>0$ be an integer. We have to showthat $\sigma (mn)>2mn$, where $\sigma (n)$ is the sum of the positive divisors of $n$.Let $d_1,\mathrm{\dots },d_k$ be the positive divisors of $n$. Thencertainly $m{d}_1,\mathrm{\dots },m{d}_k$ are distinct divisors of $mn$. The result isclear if $m=1$, so assume $m>1$. Then

$\sigma (mn)$$>$$1+{\displaystyle \sum _{i=1}^k}m{d}_i$

$>$$m{\displaystyle \sum _{i=1}^k}{d}_i$

$>$$m(2n)$

$=$$2mn.$

As a corollary, the positive abundant numbers form a semigroup.