Limit (Sequence)

analysis

Definition

Limit (Sequence)

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 $a$:

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

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.