Example 4.17.  The limit supremum of the sequence    is   ,   because if we set  ,  then for any  ,  there are only finitely many terms in the sequence larger than  .  Additionally, if L is smaller than 5, then by setting  ,  we can find infinitely many terms in the sequence larger than    (because  ).

Explore Solution 4.17.

In this case the even terms    tend to the limit 4 and the odd terms    tend to the limit 5.

The limit superior is the largest limit point of a subsequence of  .  

We can use Mathematica to graph the sequence and look for the largest limit point of a subsequence.

We see that the limit supremum of the sequence   is  .  

(c) 2006 John H. Mathews, Russell W. Howell