This is Exercise 12.12 in the Ross textbook
and I found the solution like the below, but I'm not sure how the yellow line is resulted from the above lines. Can you help me understand this solution please?
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See also: If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$. (But I don't think this should be closed as duplicate, because this one asks about a step in the proof of this.) – Martin Sleziak Aug 02 '13 at 07:58
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That is why it is always better to include more words. Consider the sequence defined by $v_n=\inf\{s_k:k>n\}$. Note that $\lim\limits_{n\to\infty}v_n=\liminf\limits_{n\to\infty}s_n$. Now, fix $N>0$, then take $M>N$. Whenever $n>M$ we will have that
$$\begin{align}\sigma_n&=\frac 1 n\left(s_1+\cdots+s_N+s_{N+1}\cdots+s_M+\cdots+s_n\right)\\&=\frac{1}{n}(s_1+\cdots+s_N)+\frac 1 n(s_{N+1}+\cdots+s_{M}+\cdots s_n)\\&\geq\frac{1}{n}(s_1+\cdots+s_N)+\frac{1}{n}(v_N+\cdots+v_N+\cdots +v_N)\\&=\frac 1 n(s_1+\cdots+s_N)+\frac{N-n}{n}v_N\end{align}$$
Since $N$ is fixed we will have, by taking $\liminf\limits_{n\to\infty}$ that $$\liminf\limits_{n\to\infty}\sigma_n\geq v_N$$ for each $N>0$. This means $$\liminf\limits_{n\to\infty}\sigma_n\geq \liminf_{n\to\infty }s_n=\lim_{N\to\infty }v_N$$
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