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Math Help - Prove that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))

  1. #1
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    Prove that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))

    Hi there, my first question here. Hope someone can give me some hints on it. Thanks!

    Let (x_{n}) be a bounded sequence. For each n \in \mathbb{N}, let y_{n}=x_{2n} and z_{n}=x_{2n-1}. Prove that
    \lim \sup {x_n} = \max (\lim \sup {y_n},\lim \sup {z_n})
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  2. #2
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    Re: Prove that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))

    To show that \limsup x_n = L (I'm using L to represent \max(\limsup y_n,\limsup z_n)), you need to show that x_n < L for all n, and for every \epsilon > 0, there is an n such that x_n > L-\epsilon. Can you see how to show each part?

    - Hollywood
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