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The theorem of BolzanoWeierstrass 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
 Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence
 Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence
 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
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
 Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016