Multiple of Perfect Number is Abundant

Theorem
Let $n$ be a perfect number.

Let $m$ be a positive integer such that $m > 1$.

Then $m n$ is abundant.

Proof
We have by definition of $\sigma$ function and perfect number that:
 * $\sigma \left({n}\right) = 2 n$

Each of the divisors of $n$ can be multiplied by $m$, and these numbers will all be divisors of $m n$.

Thus:
 * $\sigma \left({m n}\right) \ge 2 m n$

But then $1$ is still a divisors of $m n$.

So the sum of divisors of $m n$ is at least $2 m n + 1$.

Hence the result by definition of abundant.