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(1eps,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
