Let be $\displaystyle x_1 \geq x_2 \geq ... \geq x_n \geq ...$ positive numbers and $\displaystyle \sum _{n=1}^{\infty}x_n < \infty.$ Prove that $\displaystyle \lim_{n \to \infty}nx_n =0.$
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We have $\displaystyle 0\leq nx_{2n}\leq x_{n+1}+\ldots +x_{2n}=s_{2n}-s_n$ where $\displaystyle s_n:=\sum_{k=1}^n x_k$. Hence the limit of the subsequence $\displaystyle \left\{2nx_{2n}\right\}$ is $\displaystyle 0$. Now show that the limit is $\displaystyle 0$ for the subsequence $\displaystyle \left\{(2n+1)x_{2n+1}\right\}$.
The series with positive terms $\displaystyle \displaystyle \sum_{n=1}^{\infty} x_{n}$ converges and that means that there is an $\displaystyle \varepsilon >0$ and an $\displaystyle \alpha>0$ such that for n 'large enough' is...
$\displaystyle \displaystyle x_{n} < \frac{\alpha}{n^{1+\varepsilon}}$ (1)
From (1) it follows that...
$\displaystyle \displaystyle \lim_{n \rightarrow \infty} n\ x_{n} =0$ (2)
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$\displaystyle \chi$ $\displaystyle \sigma$