ThmDex – An index of mathematical definitions, results, and conjectures.
Convergent sequence in metric space has unique limit point
Formulation 0
Let $M = (X, \mathcal{T}_d, d)$ be a D1107: Metric space such that
(i) $x : \mathbb{N} \to X$ is a D336: Convergent sequence in $M$
(i) $\lim x$ is the D3000: Set of sequence limit points of $x$ in $M$
Then \begin{equation} |\lim x| = 1 \end{equation}
Proofs
Proof 0
Let $M = (X, \mathcal{T}_d, d)$ be a D1107: Metric space such that
(i) $x : \mathbb{N} \to X$ is a D336: Convergent sequence in $M$
(i) $\lim x$ is the D3000: Set of sequence limit points of $x$ in $M$
This result is a consequence of the results
(i) R498: Convergent sequence in Hausdorff space has unique limit point
(ii) R4486: Metric space is Hausdorff

$\square$