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Definition: Limits and Accumulation Points of Sequences

Let $(X,\mathcal O)$ be a topological space and let $(a_n)_{n\in\mathbb Z}$ be a sequence with $a_n\in X$ for all $n\in\mathbb Z$, and let $x\in X.$

If every open set $O\in\mathcal O$ containing $x$ contains

... all but finitely many terms of the sequence¹, then $x$ is called a limit point of the sequence $(a_n)_{n\in\mathbb Z},$ and we say that $(a_n)_{n\in\mathbb Z}$ converges to $x,$
... infinitely many terms of the sequence $(a_n)_{n\in\mathbb N}$, then $x$ is called an accumulation point of the sequence.

Notes

A sequence converges against $x$ if only finitely many sequence members are located outside any given neighborhood of $x.$
A sequence can be associated with its carrier set. Therefore, these definitions are specializations of the more general definitions of limits and $\omega$-accumulation points for sets.
Sequences do not have condensation points, since they do not have uncountable many members, by definition.

Examples: 1
Parts: 2

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References

Bibliography

Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984

Footnotes

This is equivalent to saying the for every open set $O\in\mathcal O$ containing $x$ there exists an index $N\in\mathbb N$ such that $O$ contains all sequence members $a_n$ for all $n>N.$ ↩