Notice that if is the supremum for and is supremum for then it means for and for . Thus, for . Thus, . Take the limit, .

Say that converge so for some .ii). there is equality if either or converges.

Thus, from above, we know that .

Let , this means (a result above real sequences) that there exists a convergent subsequence so that . Notice that , thus, is a subsequence of with a limit of . But the limit superior is the largest of all subsequential limits and so .

Thus, we have proved .