How to find following limit of sequence? 
How to following limit
$$\lim _{n \rightarrow \infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+\cdots+\frac{(n-1)}{n^{2}}\right)\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right). $$

My attempt:
\begin{align*}
L_n&=\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+\cdots+\frac{(n-1)}{n^{2}}\right)\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right).\\
&=\left(\frac{n-1}{2n}\right)\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right).\\
\end{align*}
By using camparison with
$$\frac{n}{2n}\le \left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right)\le \frac{n}{n+1}$$
I got $$\frac{1}{4}\le \lim_{n\to \infty}L_n\le \frac{1}{2}$$
But I need exact answer. But I could not get. Its answer given is $\ln \sqrt 2$.
Thank you very much for your valuable time.

Hint:
Rewrite the second factor as
$$\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}=\frac 1n\biggl(\frac 1{1+\dfrac1n}+\frac 1{1+\dfrac2n}+\dots+\frac 1{1+\dfrac nn}\biggr) $$
and observe this is a Riemann sum of the function $\dfrac1{1+x}$.
