Pair of convergent subsequences

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SUMMARY

In the discussion on convergent subsequences of a bounded but divergent sequence (xn), it is established that there exist two subsequences, (xnk) and (xmk), such that the limit of their difference is greater than zero. The argument begins with the fact that any bounded sequence of real numbers contains a convergent subsequence, denoted as (xnk). After removing this subsequence from the original sequence, the remaining sequence still retains boundedness and must also have a convergent subsequence, (xmk), which converges to a limit distinct from that of (xnk). This ensures that the limit of the difference |xnk - xmk| remains greater than zero.

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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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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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