ThmDex – An index of mathematical definitions, results, and conjectures.
Result R1089 on D336: Convergent sequence
Characterisation of convergent sequences in metric space
Formulation 0
Let $M = (X, \mathcal{T}_d, d)$ be a D1107: Metric space such that
(i) $x : \mathbb{N} \to X$ is a D62: Sequence
Then $x$ is a D336: Convergent sequence in $M$ if and only if \begin{equation} \exists \, a \in X : \forall \, \varepsilon > 0 : \exists \, N \in \mathbb{N} : \forall \, n \geq N : d(x_n, a) < \varepsilon \end{equation}