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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

• Informally, $a$ is an accumulation point $B,$ if there are points of $B$ which are arbitrarily close to $a.$
• This is a special case of a general topological definition of accumulation points.

Table of Contents

Examples: 1

1. Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence
2. Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence
3. Definition: Isolated Point (Real Numbers)

Mentioned in:

Examples: 1
Proofs: 2 3 4 5 6
Propositions: 7 8 9 10 11 12
Theorems: 13

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
2. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016