Superabundant Numbers are Infinite in Number

Theorem
There are infinitely many superabundant numbers.

Proof
the set $S$ of superabundant numbers is finite.

Let $m$ be the greatest element of $S$.

By definition of superabundant, $m$ has the largest abundancy index of all the elements of $S$.

Consider the integer $2 m$.

From Abundancy of Product is greater than Abundancy of Proper Factors, $2 m$ has a higher abundancy index than $m$.

There are two possibilities:


 * $(1): \quad 2 m$ is the smallest integer greater that $n$ which has a higher abundancy index than $m$.

By definition, that would make $m$ superabundant.


 * $(2) \quad$ There exists a finite set $T := \left\{ {n \in \Z: m < n < 2 m: A \left({n}\right) > A \left({m}\right)}\right\}$, where $A \left({n}\right)$ denotes the abundancy index of $n$.

The smallest element $t$ of $T$ therefore has an abundancy index greater than all smaller positive integers.

Thus by definition $t$ is superabundant.

In either case, there exists a superabundant number not in $S$.

Thus $S$ cannot contain all superabundant numbers.

But this contradicts our initial assumption that the set $S$, containing all superabundant numbers is finite.

It follows by Proof by Contradiction that $S$ is infinite.

Hence the result.