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.

Explanations

L1 human

A cluster point is a number that the sequence visits again and again, as closely as you like. A limit is a cluster point that the sequence approaches with all its late terms, not only with some of them.