Pair of convergent subsequences

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Let (xn) be a bounded sequence that diverges. Show that there is a pair of convergent subsequences (xnk) and (xmk), so that

[tex]
lim_{k\rightarrow\infty}
[/tex] [tex]\left|x_{nk} - x_{mk}\right| > 0[/tex]​
 

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  • #2
HallsofIvy
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Any bounded sequence of real numbers has a convergent subsequence- let that be (xnk). Removing those from the sequence gives a new sequence that is still bounded. It must also have a convergent subsequence. Further, there must exist a subsequence that converges to something other than the limit of (xnk) (if all convergent subsequences converged to the same thing, the sequence itself would be convergent) . Let such a subsequence be (xmk).
 

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