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 $n\ge N.$

  1. Lemma: Criteria for Convergent Sequences

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