Sequences, series, summability Definition

Cluster point of a sequence

Definition

Let $(a_n)$ be a real sequence and let $c$ be a real number. The number $c$ is a cluster point of $(a_n)$ if the following holds for each $\varepsilon > 0$ and each natural number $N$ . There is a natural number $n \ge N$ such that $|a_n - c| < \varepsilon$ .

Remarks

Equivalently, for each $\varepsilon > 0$ there are infinitely many indices $n$ with $|a_n - c| < \varepsilon$ . In $\mathbb{R}$ , the number $c$ is a cluster point of $(a_n)$ if and only if some subsequence of $(a_n)$ converges to $c$ . The sequence $(-1)^n$ has the two cluster points $1$ and $-1$ , and no limit. For a bounded sequence, the limit superior is the largest cluster point and the limit inferior is the smallest.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Sequence definition
Real sequence definition
Subsequence definition
Cluster point of a sequence definition

Tag	`0RCKQN1E09`
URI	`m2://0RCKQN1E09`
Aliases	`accumulation-point-of-a-sequence`
Kind	definition
Writing	W0 stub
MSC	40A05
Defines	Cluster point of a sequence
Uses	Real sequence, Subsequence

Cluster point of a sequence

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here