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applicability: $\mathbb {R}$

Definition: Convergent Real Sequence

A convergent real sequence is a real sequence $$(x_n)_{n\in\mathbb N}$$, which is convergent in the metric space of real numbers $$(\mathbb R,|~|)$$. In other words, $$(x_n)_{n\in\mathbb N}$$ is convergent to the number $$x\in\mathbb R$$, i.e. for each $$\epsilon > 0$$ there exists an $$N\in\mathbb N$$ with
$| x_n-x | < \epsilon\quad\quad \text{ for all }n\ge N.$

If $$(x_n)_{n\in\mathbb N}$$ is convergent to the number $$x\in\mathbb R$$, we write
$\lim_{n\to\infty} x_n=x.$

In this case, the convergent sequence is also said to be convergent to the number (or tending to the number) $x$.

| | | | | created: 2014-02-20 22:39:58 | modified: 2020-07-09 05:00:28 | by: bookofproofs | references: [581], [582]

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[582] Forster Otto: “Analysis 2, Differentialrechnung im $$\mathbb R^n$$, Gewöhnliche Differentialgleichungen”, Vieweg Studium, 1984