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Prove: Let $(a_n)_{n=1}^\infty$ be a monotonously falling sequence of non-negative real numbers, so that $\sum a_n$ is convergent. Then $(na_n)$ is a sequence converging to 0.

Well, no idea. All help is greatly appreciated!


marked as duplicate by Clarinetist, mrf, zipirovich, Lee Mosher, Rolf Hoyer Dec 5 '17 at 23:51

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Because $a_n$ is a non-negative real number, let's say that $c$ is the smallest number of the $a_n$ sequence. That means that $\sum_{n=1}^\infty (n *a_n) \geq \sum_{n=1}^\infty (c)$. Well, if it is that $c>0$ then that series would be divergent. It is not, thus $c=0$, so the series will converge, and may only converge, to $0$.

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    $\begingroup$ There is not a smallest such number. What you want instead is the infimum. Your estimate does not work, though, because no one is asking you to prove that $\sum_n n a_n $ converges. $\endgroup$ – Andrés E. Caicedo Dec 5 '17 at 13:48

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