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Math Help - Sequence

  1. #1
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    Sequence

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

    \sigma_K=\frac{s_1+\cdots+s_K}{K}=\frac{s_1+\cdots  +s_n}{K}+\frac{s_{n+1}+\cdots+s_K}{K}
    >\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 \forall\epsilon>0 \exists K_0 s.t. \sigma_K>s-1.5\epsilon for all K\ge K_0, which implies \lim\inf \sigma_n\ge s

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