# Definition: Convergent Sequences and Limits

Let $$(X,d)$$ be a metric space. The sequence $$(a_n)_{n\in\mathbb N}$$ with points $a_n\in X$ is called convergent to the point $$a\in X$$, if and only if for each neighborhood $$B(a,\epsilon)$$ there exists an $$N\in\mathbb N$$ with $a_n\in B(a,\epsilon)$ for all $n\ge N$.

Alternative formulation: The sequence $$(a_n)_{n\in\mathbb N}$$ is convergent to $$a\in X$$, if and only if all but finitely many sequence members $a_n$ lie in the neighborhood $$B(a,\epsilon)$$, no matter how small $$\epsilon > 0$$ is.

The point $$a\in X$$ is called the limit of the sequence $$(a_n)$$ and is denoted by $$\lim_{n\rightarrow\infty} a_n=a.$$

Note that if $$(X,d)$$ is a normed vector space $$(X,d)=(X,\|\|)$$ , this definition is equivalent to the following definition: For each $$\epsilon > 0$$ there exists an $$N\in\mathbb N$$ with $\| a_n-a \| < \epsilon$ for all $n\ge N.$

Corollaries: 1
Definitions: 2 3 4 5 6 7
Lemmas: 8 9
Proofs: 10 11 12 13 14 15 16 17 18 19
Propositions: 20 21 22 23
Theorems: 24

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### References

#### Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im $$\mathbb R^n$$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984