$\lim_{n\to\infty} \sum_{r=1}^n \frac{n}{n+r} - n\log 2$

Consider the following two sequences of reals $$a_n=\frac{1}{n+1}+\cdots +\frac{1}{2n}$$ and $$b_n=\frac{1}{n}$$ How would one prove that $$\lim_{n\to\infty }\frac {a_n-\log 2}{b_n}$$ is finite and non-zero? Actually this is part of one of my old questions: Rate of convergent of real Sequence. But I am unable to to prove that $$\lim_{n\to\infty} \frac{a_n - \log 2}{b_n} = \lim_{n\to\infty} \sum_{r=1}^n \frac{n}{n+r} - n\log 2 = -\frac{1}{4} \ne 0, \pm\infty$$ I tried it but didn't get the limit. Thank you .