limit superior

When a sequence refuses to converge, it may still have a well-defined 'eventual high-water mark'. The limit superior is the largest value the sequence keeps approaching infinitely often — the ceiling that its terms repeatedly rise to (or toward), even if they keep falling back down. It exists for every real sequence, where an ordinary limit may not.

There are two equivalent definitions. First, limsup a_n is the largest subsequential limit — the biggest value attainable as a limit of some subsequence (allowing +infinity). Second, and most useful for computation: limsup a_n = lim_{n->inf} ( sup_{k >= n} a_k ). The inner supremum sup_{k >= n} a_k is the highest value in the tail from index n on; as n grows the tail shrinks, so these suprema form a decreasing sequence whose limit is the limsup.

The limit superior always exists in the extended reals [-infinity, +infinity], which is why it is so handy: every sequence has one. A sequence converges to L (finite) if and only if its limsup and liminf are both equal to L. So limsup and liminf together pin down convergence and, when they disagree, measure exactly how far a sequence is from settling down.