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Thread: Sequence

  1. #1
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    Sequence

    Let $\displaystyle (s_n)$ be a sequence of nonnegative numbers and for each n define $\displaystyle \sigma_n = \frac{s_1+s_2+...+s_n}{n}$.
    show that $\displaystyle \lim \inf s_n \le \lim \inf \sigma_n \le \lim \sup \sigma_n \le \lim \sup s_n$
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  2. #2
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    let $\displaystyle \lim\inf s_n=s$. this means that $\displaystyle \forall\epsilon>0 \exists N$ s.t. $\displaystyle s_n>s-\epsilon$ for all $\displaystyle n\ge N$. then

    $\displaystyle \sigma_K=\frac{s_1+\cdots+s_K}{K}=\frac{s_1+\cdots +s_n}{K}+\frac{s_{n+1}+\cdots+s_K}{K}$
    $\displaystyle >\frac{s_1+\cdots+s_n}{K}+\frac{(s-\epsilon)(K-N)}{K}>s-1.5\epsilon$ for all large enough K. we showed that $\displaystyle \forall\epsilon>0 \exists K_0$ s.t. $\displaystyle \sigma_K>s-1.5\epsilon$ for all $\displaystyle K\ge K_0$, which implies $\displaystyle \lim\inf \sigma_n\ge s$

    middle inequality is trivial, the right inequlaity is done in exactly the same way
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