If $\lim_{n \rightarrow \infty }s_n = s$, is it true that $\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n s_k =s $ as well?

Question

If $\{s_n\}$ is a sequence of positive real numbers such that $\lim_{n \rightarrow \infty }s_n = s$, is it true that $\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n s_k =s $ as well?

Answer 1

This is true. Let $\varepsilon>0$ be given, choose $N>0$ such that $n\geq N$ implies that $|s_n-s|<\varepsilon$ then for $M>N$

$$\left|\sum_{k=1}^{M}\frac{s_k}{M}-s\right|=\left|\sum_{k=1}^{M}\frac{s_k-M\cdot s}{M}\right|=\left|\frac{1}{M}\sum_{k=1}^{M}(s_k-s)\right|\leq \frac{1}{M}\left|\sum_{k=1}^{N}(s_k-s)\right|+\frac{1}{M}\sum_{k=N+1}^{M}\varepsilon$$

Do you see how one could proceed from this?