Let $n$ be a positive integer and $j=1,2,\ldots, n;$ then I found in an article saying that by using approximations by integrals it is easy to show that $$ \frac{j}{n+1}<1-e^{-(\frac{1}{n}+\cdots + \frac{1}{n-j+1})}<\frac{j}{n+\frac{1}{2}} $$
But I do not see that this is easy and don't know how to use the integral approximations. Any help would be great to proceed further.
Thanks!