OIMU 2007 Problem 7

The height of a positive integer is defined as being the fraction $\frac{s(a)}{a}$ , where $s(a)$ is the sum of all the positive divisors of $a$ . Show that for every pair of positive integers $N,k$ there is a positive integer $b$ such that the height of each of $b,b+1,\cdots,b+k$ is greater than $N$ .