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Let $(s_n)$ be a sequence of real numbers and $(s_n)\geq -C$ for some $C\geq 0.$

I wonder if the sequence of Cesaro means of $(s_n)$ $$\sigma_n(s)=\frac{1}{n+1}\sum_{k=0}^{n}s_k$$ is also one-sided bounded i.e$$\sigma_n(s)\geq-C.$$

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    $\begingroup$ That's fairly obvious, dont you think? You can add inequalities, and divide by a positive number. $\endgroup$
    – Bernard
    Nov 21, 2015 at 21:45
  • $\begingroup$ I would be grateful if you give more clear hint. $\endgroup$
    – Raio
    Nov 21, 2015 at 23:20
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    $\begingroup$ I gave the details in my answer. $\endgroup$
    – Bernard
    Nov 21, 2015 at 23:27

1 Answer 1

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Add the inequalities: \begin{align*} s_0&\ge -C\\ &\vdots\\s_n&\ge -C \end{align*} to get $\;\displaystyle\sum_{k=0}^ns_k\ge (n+1)(-C)$, whence $\;\displaystyle\frac1{n+1}\sum_{k=0}^ns_k\ge -C$.

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