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The theorem of Bolzano-Weierstrass motivates the following definition:

Definition: Accumulation Point (Real Numbers)

A real number \(a\) is called an accumulation point of a real sequence \((a_n)_{n\in\mathbb N}\), if it contains a subsequence \((a_{n_k})_{k\in\mathbb N}\) that is convergent to \(a\).

Notes

| | | | | created: 2014-02-20 23:23:56 | modified: 2020-07-09 05:27:36 | by: bookofproofs | references: [581], [6823]

1.Example: Examples of Accumulation Points

2.Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence

3.Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence

4.Definition: Isolated Point (Real Numbers)

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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983

[6823] Kane, Jonathan: “Writing Proofs in Analysis”, Springer, 2016