Firstly, I don't think such a sequence can possibly exist.
Secondly, I'm not sure what your sequence is... Is n fixed? Is j fixed?
I'm trying to construct an infinite sequence that converges to zero but has a positive (nonzero) limit supremum. Here's what I'm attempting to do:
Let a_j = 1 if j = n+1 and 0 otherwise. Then the limit supremum equals (lim n -> inf) sup {a_n, a_n+1, a_n+2, ...}
= (lim n -> inf) sup {0, 1, 0, 0, ...}
= 1
Am I allowed to do that, or is that "cheating"? If so, what's a workaround that would suffice?