Limit Point

analysis

Definition

Limit Point

A number is called limit point of sequence if infinitely many sequence elements lie in every epsilon neighbourhood of .

Example: In the sequence below, the limit point is .

Different Limit Points

Let and be two different limit points of sequence , then for it holds that . Hence, it is impossible for almost all to lie in both and .

Uniqueness of Limits

From this consideration it follows the uniqueness of the limit: since almost all terms of the sequence lie in every neighbourhood of the limit, a convergent sequence can have only one limit point. In the case of convergence, it therefore holds that:

Examples

Find Limit Points of

Let be a sequence:

We can plot this sequence as a simple number line:

For , we can now choose an arbitrary epsilon neighbourhood with and plot it:

Since we have an infinite number of occurrences of for any , we know that is a limit point of that sequence.

Analogously, the same can be observed for :

Thus, we can conclude that and are limit points of sequence .

Further, we can also observe that this sequence does not have a limit since there is an infinite number of points in the left epsilon neighbourhood and an infinite number of points in the right epsilon neighbourhood. However, a limit postulates that there is only a finite number of exceptions that lie not inside the epsilon neighbourhood. Therefore, whenever we have two limit points, a sequence does not have a limit. This also implies that there can’t be two limits in a single sequence since every limit is also a limit point.