Abundancy Index of Product is greater than Abundancy Index of Proper Factors

Theorem
Let $n \in \Z_{>0}$ be a composite number such that $n = r s$, where $r, s \in \Z_{>1}$.

Then:
 * $\dfrac {\sigma \left({n}\right)} n > \dfrac {\sigma \left({r}\right)} r$

and consequently also:
 * $\dfrac {\sigma \left({n}\right)} n > \dfrac {\sigma \left({s}\right)} s$

where $\sigma \left({n}\right)$ denotes the $\sigma$ function of $n$.

That is, the abundancy index of a composite number is strictly greater than the abundancy index of its divisors.

Proof
Consider the divisors of $r$.

Let $d \mathrel \backslash r$, where $\backslash$ indicates divisibility.

We have that:
 * $d \mathrel \backslash n$

and also that:
 * $d s \mathrel \backslash n$

Thus:

Similarly for $s$.