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Thread: [SOLVED] Another lim inf/lim sup proof

  1. #1
    Senior Member Pinkk's Avatar
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    [SOLVED] Another lim inf/lim sup proof

    Let $\displaystyle (s_{n})$ be a sequence of nonnegative numbers, and for each n define $\displaystyle a_{n} = \frac{1}{n}(s_{1} +s_{2}+...+ s_{n})$. Show that $\displaystyle \lim \,inf \,s_{n} \le \lim \,inf \,a_{n} \le \lim \,sup \,a_{n} \le \lim \,sup \,s_{n}$. Also show that if $\displaystyle \lim s_{n}$ exists, then $\displaystyle \lim a_{n}$ exists and $\displaystyle \lim a_{n} = \lim s_{n}$.

    For this one I am completely stuck. Any help would be appreciated.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    Let $\displaystyle (s_{n})$ be a sequence of nonnegative numbers, and for each n define $\displaystyle a_{n} = \frac{1}{n}(s_{1} +s_{2}+...+ s_{n})$. Show that $\displaystyle \lim \,inf \,s_{n} \le \lim \,inf \,a_{n} \le \lim \,sup \,a_{n} \le \lim \,sup \,s_{n}$. Also show that if $\displaystyle \lim s_{n}$ exists, then $\displaystyle \lim a_{n}$ exists and $\displaystyle \lim a_{n} = \lim s_{n}$.

    For this one I am completely stuck. Any help would be appreciated.
    What is the definition of $\displaystyle \limsup$?
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  3. #3
    Senior Member Pinkk's Avatar
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    $\displaystyle lim sup = \lim_{N\to \infty}\, sup\{s_{n} : n > N\}$
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  4. #4
    Senior Member Pinkk's Avatar
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    Any suggestions?
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  5. #5
    Senior Member Pinkk's Avatar
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    Apparently, for part of the proof I need to show that $\displaystyle M > N$ implies $\displaystyle sup\{a_{n} : n > M\} \le \frac{1}{M}(s_{1}+s_{2}+...+s_{N}) + sup\{s_{n} : n > N\}$ (and I don't even how to show that or why that's true...
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  6. #6
    Senior Member Pinkk's Avatar
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    Sorry for all the consecutive posts, but I am still absolutely stuck on this.
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