Example of sequence which has every point of [0,1] as limit point if it exists

Question

I wanted to find
1)Example of the sequence which has every point of [0,1] as limit point if it exists
2)Example of the sequence which has every rational point of [0,1] as limit point if it exists
Sorry, I thought but I did not get example.Any Help will be appreciated

Answer 1

$\sin^2 n$ is as good a sequence as any.

Answer 2

Concerning the second question, if a sequence of elements of $[0,1]$ is such that every rational in $[0,1]$ is a limit point, then every element of $[0,1]$ is a limit point, since the rationals are dense. So, it is not possible that only the rationals are limit points.