Examining Properties of a Bounded Sequence

We first examine some examples regarding the liminf and limsup of a bounded sequence, and we will discover theorems regarding them after.

Def: A bounded sequence has the property that for all , we can define the following:

Example 1

Take the example of

In this case, we observe that , so we can see that seems to converge.

Example 2

Take the example of

Sequence Limit

In this case, we can see that seems to diverge.

Lemma: Boundedness and Infimums and Supremums

Based on our observations above,