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.

| | | | | 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)