For a given sequence $a_1,a_2,\ldots,a_n$ if $\lim\limits_{n\to \infty}a_n=a$, then $\lim\limits_{n\to \infty}\dfrac{1}{\ln(n)}\sum\limits_{r=1}^{n}\dfrac{a_r}{r}$ is:
(A)zero $\hspace{150pt}$ (B)a
(C)$\dfrac{a}{2}$ $\hspace{157pt}$ (D)None of these
My approach: I'm trying to bring this limit into this form: $\lim\limits_{n\to \infty}\dfrac{1}{n}\sum\limits_{r=1}^{n}f\left(\dfrac{r}{n}\right)$ as, $$\lim\limits_{n\to \infty}\dfrac{1}{n}\sum\limits_{r=1}^{n}f\left(\dfrac{r}{n}\right)=\int_{0}^{1}f(x)\ dx$$ so here it is what I'm doing: \begin{align*} \lim\limits_{n\to \infty}\dfrac{1}{\ln(n)}\sum\limits_{r=1}^{n}\dfrac{a_r}{r} &=\lim\limits_{n\to \infty}\dfrac{1}{n}\cdot\dfrac{n}{\ln(n)}\sum\limits_{r=1}^{n}\dfrac{a_r}{r}\\ &=\lim\limits_{n\to \infty}\dfrac{1}{n}\sum\limits_{r=1}^{n}\dfrac{a_r}{r/n}\cdot\dfrac{1}{\ln(n)} \end{align*} but I can't make the limit in the above form, is there any way of doing it?