Limit

analysis

Definition

Grenzwert

Limit

A real number $a$ is called limit of sequence $(a_n{)}_{n\ge 0}$ if almost all sequence elements $a_n$ lie within every epsilon neighbourhood of $a$:

$$\forall \epsilon >0:\exists N(\epsilon )\in \mathbb{N}:\forall n\in N(\epsilon ):|a_n-a|<\epsilon$$

A number $a$ is thus the limit of a sequence if the following holds: Given any distance $\epsilon$, an index $N$ can be found such that from this index $N$ onward, all sequence elements have a from/to $a$ that is smaller than $\epsilon$. Such an $N$ must exist for every $\epsilon$, no matter how small $\epsilon$ is chosen.

If there exists an epsilon neighbourhood of $a$ that does not contain infinitely many sequence elements, then $a$ cannot be the limit, then $a$ cannot be the limit of that sequence.

Every limit is a limit point of that sequence.

Uniqueness

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:

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