Let $x_{n}$ be a monotonously decreasing sequence converging to $0$. The series $\sum x_{n}$ is convergent. Prove that $nx_{n}$ is also convergent to $0$.
I‘m kinda stuck here. Help would be appreciated.
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7Does this answer your question? [If $(a_n)\subset0,\infty)$ is non-increasing and $\sum a_n<\infty$, then $\lim{n a_n} = 0$ – Jonah May 04 '21 at 17:55
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Using Cauchy criterion, one has \begin{align*} \left|\sum_{k=n}^{2n}x_{k}\right|<\epsilon, \end{align*} but then \begin{align*} \left|\sum_{k=n}^{2n}x_{k}\right|=\sum_{k=n}^{2n}x_{k}\geq\sum_{k=n}^{2n}x_{2n}\geq nx_{2n}=(1/2)(2n)x_{2n}. \end{align*} Similarly, $(2n+1)x_{2n+1}$ is arbitrarily small.
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