Pair of convergent subsequences

[bolzano07]

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

<br /> lim_{k\rightarrow\infty}<br /> \left|x_{nk} - x_{mk}\right| &gt; 0​

 

[HallsofIvy]

Any bounded sequence of real numbers has a convergent subsequence- let that be (x_nk). 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 (x_nk) (if all convergent subsequences converged to the same thing, the sequence itself would be convergent) . Let such a subsequence be (x_mk).