Let be a bounded sequence that diverges. Show that there is a pair of convergent subsequences and so that
> 0
The equation can equivalently be written since these limits are assumed to exist.
I guess you know that a bounded sequence has a convergent subsequence.
Then choose a convergent subsequence , with some limit . Using the fact that initial sequence does not converge, you can find and a subsequence such that for all (I let you justify that); then extract a convergent subsequence from and conclude.