0 and 1 seem to me to be subsequential limits. 1 is greater than 0. No?
Any convergent subsequence much have its terms sufficiently close to one another (Cauchy sequence). However, all these terms are composed of integers so the only way to get a convergent subsequence is that it is eventually constant. Thus, convergent subsequences must eventually be composed of entirely 0's or entirely 1's. Thus, 0 and 1 are the subsequential limits. Therefore, the limit superior is 1 while the limit inferior is 0.