1) Is it Possible to find a sequence for which the set of sub-sequential limit points is [0,1]?
2)Is any closed set in R,the set of sub-sequential limit points of a sequence?
a) Consider the sequence {x_n} such that there is a one to one correspondence between the elements of the sequence and the rational numbers in [0,1]. Then the sequence has sub-sequential limits [0,1].
b) Yes. Any finite set is closed. So try to find a sequence with finite no. of limit points.
There exist a bijection between any two countable sets:
If A is countable then, by definition, there exist a bijection f:N-> A.
If B is countable then, by definition, there exist a bijection g:N-> B.
The function A->B, $\displaystyle g\circ f^{-1}$ is a bijection from A to B.