Limit Point is Limit of Convergent Sequence/Proof 2
Theorem
Let $M = \left({X, d}\right)$ be a metric space.
Let $E \subseteq X$ be a subset of $X$.
Let $p$ be a limit point of $E$.
Then there exists a sequence $\left\langle{x_n}\right\rangle \subseteq E$ which converges to a limit:
- $\displaystyle \lim_{n \mathop \to \infty} x_n = p$
where $\displaystyle \lim_{n \mathop \to \infty} x_n$ is the limit of the sequence $\left\langle{x_n}\right\rangle$.
Proof
From Metric Induces Topology, a metric space induces a topological space.
Then Equivalence of Definitions of Limit Point can be applied.
$\blacksquare$
Use a link to the definition rather than the equivalence proof.
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- 1953: Walter Rudin: Principles of Mathematical Analysis: $3.2d$