Convergence

**yusukered07** · March 13th 2010, 11:53 PM

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

$lim_{k\rightarrow{\infty}}\mid{x_{nk}} - {x_{mk}}\mid$ > 0

**Laurent** · March 14th 2010, 05:36 AM

Originally Posted by yusukered07

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

$lim_{k\rightarrow{\infty}}\mid{x_{nk}} - {x_{mk}}\mid$ > 0

The equation can equivalently be written $\lim_k x_{n_k}\neq \lim_k x_{m_k}$ since these limits are assumed to exist.

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

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

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