Suppose that $a_n > 0$ and that, for n sufficiently large, $$\frac{a_{n+1}}{a_n}\leq 1- \frac{c}{n}$$ for $c>1$
Prove that the series $\displaystyle \sum_{n=1}^{\infty}a_n$ converges.
Hint supplied : Prove the inequality $log(1-x)<-x$ for $0<x<1$
I'm not entirely sure how to approach this, the ratio test fails and I'm trying to show the series is Cauchy but to no avail.