Example 4.18.  The limit supremum of the sequence   is   ,   because if we set  ,  then for any  ,  there are only finitely many terms (actually, there are none) in the sequence larger than  .  Additionally, if L is smaller than 3, then by setting    we can find infinitely many terms in the sequence larger than  ,  because  ,  as the following calculation shows:  

            .  

Explore Solution 4.18.

In this case there are only three different values for the terms in the sequence.

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