Can Two Subsequences Converge to 0 and 1 Respectively?

[playa007]

Homework Statement

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.

Homework Equations

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.

 

[lurflurf]

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