Nonexistence of Odd Perfect Numbers

Let $\sigma(n)$ denote the sum-of-divisors function $\sigma(n)=\sum_{d \mid n} d$. We also define $I(n) = \frac{\sigma(n)}{n}$ as the abundancy index of $n$. An integer $n$ is perfect if $I(n)=2$. It is unknown whether any odd perfect number exists or not. In this note, using the properties of the abundancy index function, we prove the non-existence of odd perfect numbers.