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,...

S is countable.  If the set of distinct elements in S is the same as the set of distinct subsequental limits, then there must be a countable set of distinct subsequential limits.  This rules out many sequences S.  For instance, S cannot contain all rational numbers.

Posted by Steve Herman on 2006-12-14 16:54:50