1. The problem statement, all variables and given/known data Suppose {a_n} is a bounded sequence who's set of all subsequential limits points is {0,1}. Prove that there exists two subsequences, such that: one subsequence converges to 1 while the other converges to 0, and each a_n belongs to exactly one of these subsequences. 2. Relevant equations 3. The attempt at a solution Well, it's clear that at the limit points 0 and 1; there is a subsequence that that converges to it. I'm not quite sure about how to prove that each a_n belongs to exactly one of these subsequences or how to apply the bounded property of {a_n} into this question.
This will depend slightly on the definition you are using. Suppose {a_n} is a bounded sequence who's set of all subsequential limits points is {0,1} Suppose epps is a positive real number 0 and one are limit points so it is known that (-eps,eps)U(1-eps,1+eps) Contains all but a finite number of the a_n now the sequence can be easily partitioned by 1/2 one subsequence if a_n<=1/2 another if a_n>1/2