Show that $\frac{n}{\sigma(n)} > (1-\frac{1}{p_1})(1-\frac{1}{p_2})\cdots(1-\frac{1}{p_r})$ If $n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}$ is the prime factorization of $n>1$ then show that :

$$1>\frac{n}{ \sigma (n)} > \left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right)\cdots\cdots\left(1-\frac{1}{p_r}\right)$$

I have solved the $1^\text{st}$ inequality($1>\frac{n}{ \sigma (n)}$) and tried some manipulations on the right hand side of the $2^\text{nd}$ inequality but can't get much further.Please help.

Let $S$ be the set of products of powers of the prime divisors of $n$ and $n/S$ the set of ratios $(n/s) : s \in S$.
$\frac{n}{\Pi (1 - \frac {1}{p_i})} > \sigma(n) \quad $ is the statement that the sum of all rational numbers in $n/S$ is larger than the sum of all integers in the same set.
