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?