For a given sequence S=x₁,x₂,x₃,.. of real numbers, x is called a subsequential limit of S, if x is the limit of a subsequence of S. For example, the sequence 1+2^-1,2+2^-1,1+2^-2,2+2^-2,.. has the subsequential limits 1 and 2.

Let S be a sequence so that the set of its subsequential limits is identical to the set of sequence elements. Prove that S contains a subsequence of the form x,x,x,...

(In reply to Observation 2 by Steve Herman)

Excellent observation, Steve. This is in fact the key for the proof. If one tries to construct an example of a self-limited sequence without a constant subsequence, one will always end up with a non-countable set. Remains to be proven that this must always be the case.

Posted by JLo on 2006-12-21 13:45:25