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:

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