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Thread: Convergence

  1. #1
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    Convergence

    Let $\displaystyle (x_n)$ be a bounded sequence that diverges. Show that there is a pair of convergent subsequences $\displaystyle (x_{nk})$ and $\displaystyle (x_{mk})$ so that

    $\displaystyle lim_{k\rightarrow{\infty}}\mid{x_{nk}} - {x_{mk}}\mid$ > 0
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  2. #2
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    Quote Originally Posted by yusukered07 View Post
    Let $\displaystyle (x_n)$ be a bounded sequence that diverges. Show that there is a pair of convergent subsequences $\displaystyle (x_{nk})$ and $\displaystyle (x_{mk})$ so that

    $\displaystyle lim_{k\rightarrow{\infty}}\mid{x_{nk}} - {x_{mk}}\mid$ > 0
    The equation can equivalently be written $\displaystyle \lim_k x_{n_k}\neq \lim_k x_{m_k}$ since these limits are assumed to exist.

    I guess you know that a bounded sequence has a convergent subsequence.

    Then choose a convergent subsequence $\displaystyle (x_{n_k})_k$, with some limit $\displaystyle \ell$. Using the fact that initial sequence does not converge, you can find $\displaystyle \epsilon_0$ and a subsequence $\displaystyle (x_{m_k})_k$ such that $\displaystyle |x_{m_k}-\ell|>\epsilon_0$ for all $\displaystyle k$ (I let you justify that); then extract a convergent subsequence from $\displaystyle (x_{m_k})_k$ and conclude.
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