We exclude the case of . Let , then there is a subsequence of that converges to . Since , we have . Both and has limit as (the former is a subsequence of , the latter is a subsequence of which is monotonically increasing and therefore has as its limit), so the left side approachesa+bas . Supposecis a subsequential limit of (may be ) which, as subsequence of subsequence, is also a subsequence of , so which is the supremum of the set of all subsequential limits (including possibly ).