Let a sequence of real numbers $(s_n)$ is one-sided bounded. Is the Cesaro mean of the sequence $(s_n)$ also one-sided bounded?

Question

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$$\sigma_n(s)\geq-C.$$

Answer 1

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 $\;\displaystyle\frac1{n+1}\sum_{k=0}^ns_k\ge -C$.