The theorem of Bolzano-Weierstrass motivates the following definition:

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 \(a\).

- 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.

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

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

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