Evaluate $\lim\big(\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2n}\big)$ using sequential methods 
Evaluate $$\lim\Big(\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2n}\Big)$$ using sequential methods.

Of course: $$\frac{1}{n}\bigg(\frac{1}{1+1/n}+\frac{1}{1+2/n}+\ldots+\frac{1}{1+n/n}\bigg) \rightarrow \int_{0}^{1}\frac{dx}{1+x}=\ln 2$$
but by using only sequences, I don't know where to start from. I thought of the squeeze theorem, but i am not sure how to get $\ln 2$ on the way. 
Thanks in advance.

Using the fact that $\ln(1 + x) \leq x$, we have
$$
\ln(k + 1)  - \ln k \leq \frac{1}{k} \leq \ln{k} - \ln(k - 1),
$$
hence
$$
\ln \frac{2n + 1}{n + 1} \leq \sum_{k = {n + 1}}^{2n} \frac{1}{k} \leq \ln \frac{2n}{n} = \ln2,
$$
and the squeeze theorem will give the result.
