Limit Point

analysis

Definition

Häufungspunkt

Limit Point

A number $a$ is called limit point of sequence $(a_n{)}_{n\ge 0}$ if infinitely many sequence elements lie in every epsilon neighbourhood of $a$.

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

Different Limit Points

Let $a$ and $b$ be two different limit points of sequence $(a_n{)}_{n{\ge }_0}$, then for $\epsilon <\frac{1}{2}|a-b|$ it holds that $U_{\epsilon }(a)\cap U_{\epsilon }(b)=\varnothing$. Hence, it is impossible for almost all $a_n$ to lie in both $U_{\epsilon }(a)$ and $U_{\epsilon }(b)$.

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:

$$\underset{n\to \infty }{\lim }{a}_n=\lim \underset{n\to \infty }{\sup }{a}_n=\lim \underset{n\to \infty }{\inf }{a}_n$$

Examples

Find Limit Points of $((-1{)}^n{)}_{n\ge 0}$

Let $((-1{)}^n{)}_{n\ge 0}$ be a sequence:

$$(-1{)}^n=(1,-1,1,-1,\dots )$$

We can plot this sequence as a simple number line:

For $-1$, we can now choose an arbitrary epsilon neighbourhood with $\epsilon >0$ and plot it:

Since we have an infinite number of occurrences of $-1$ for any $\epsilon >0$, we know that $-1$ is a limit point of that sequence.

Analogously, the same can be observed for $1$:

Thus, we can conclude that $1$ and $-1$ are limit points of sequence $((-1{)}^n{)}_{n\ge 0}$.

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.