Here is a result that you should know.
Lemmon: If are convergent sequences and then where is the limit of and is the limit of .
Now we can prove that . The most important thing here is to understand what means. It means the limit of the superior sequence, i.e. *. Now . Since bounded sequences always have limit superiors it means these superior sequences have limits. So by the lemmon: . Thus, .
*)Example. Say . Then , , , .... So , , .... So in general the -th term in the superior sequence is: , thus the limit is . This means .