Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space

Let the points \(x_n\) of a sequence \((x_n)_{n\in\mathbb N}\) be contained in the metric space \((X,d)\), i.e. \(x_n\in X\), for all \(n\in\mathbb N\). Let this sequence be convergent and let \(X\) also contain its limit

\[x=\lim_{n\to\infty} x_n,\] i.e. \(x\in X\). Then the subset $U\subset X$ defined by \[U:=\{x_n:~n\in\mathbb N\},\] i.e. the subset consisting of all sequence members \(x_n\), but which does not contain the limit $x$, is not compact.

Proofs: 1

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