# Pair of convergent subsequences

1. Mar 14, 2010

### bolzano07

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

$$lim_{k\rightarrow\infty}$$ $$\left|x_{nk} - x_{mk}\right| > 0$$​

2. Mar 14, 2010

### HallsofIvy

Re: Convergence

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).