Is there a sequence of real numbers such that the set of its subsequential limits is $[0,1]$? 
Is there a sequence of real numbers such that the set of its subsequential limits is $[0,1]$?

I've been considering the sequence:
$$1,~~\frac12,~~\frac22,~~\frac32,~~\frac13,~~\frac23,~~\frac33,~~\frac43,~\ldots~,~~\frac1n,~~\frac2n,~~\frac3n,~\ldots~,~~\frac{n+1}{n},~\ldots$$
Does it satisfy the condition?

That does indeed work.
Another option is to let $a_n=\vert\sin(n)\vert$ - this is a good exercise.
Or, we can pick an enumeration of $\mathbb{Q}\cap [0, 1]$.
