How to prove the following convergence proposition [duplicate]

Suppose that $$\sum\limits_{n=1}^{\infty}a_n$$converges.Prove that if $$b_n$$ monotone increasing to infinity and $$\sum\limits_{n=1}^{\infty}a_nb_n$$converges,then $$b_n\sum\limits_{k=n}^{\infty}a_k\to0(n\to\infty)$$.I made the following attempts. Let $$R_n=\sum\limits_{k=n}^{\infty}a_k$$,so $$\lim\limits_{n\to\infty}R_n=0$$.then we know that $$\{\frac{1}{b_n}\}$$ monotonic decreasing and $$\frac{1}{b_n}\to0(n\to\infty)$$.So we have $$\sum\limits_{k=n}^{\infty}\dfrac{a_k}{b_k}=\sum\limits_{k=n}^{\infty}\dfrac{1}{b_k}(R_k-R_{k+1})\implies\left|\sum\limits_{k=n}^{\infty}\dfrac{a_k}{b_k}\right|\leqslant\sum\limits_{k=n}^{\infty}\dfrac{1}{b_k}|R_k-R_{k+1}|\leqslant\dfrac{1}{b_n}\sum\limits_{k=n}^{\infty}|R_k-R_{k+1}|\leqslant\dfrac{1}{b_n}R_n.$$ So we can get it $$b_n\sum\limits_{k=n}^{\infty}\dfrac{a_k}{b_k}\to0(n\to\infty)$$.I'd like to know how to proceed. Thank you.

• Are the $a_n$ non-negative? Then $b_n \sum_{k=n}^\infty a_k \le \sum_{k=n}^\infty a_k b_k \to 0$. Mar 25 '21 at 12:28
• @ Martin R There is no explanation in the title, so... Mar 25 '21 at 12:30
• Check this: math.stackexchange.com/q/1164031/42969 Mar 25 '21 at 14:26
• @Martin R God,How do you find this problem...I probably looked at it and it didn't feel right. I thought, if we can prove it$b_n\sum\limits_{k=1}^{n}a_n\to0$.Does that mean $b_n\sum\limits_{k=n}^{\infty}a_k\to0(n\to\infty)$ holds? Mar 25 '21 at 14:33
• approach0.xyz/search/… Mar 25 '21 at 14:39