Limit points are limits of convergent subsequences in a first-countable space

Let $𝑋$ be a first-countable topological space and $(𝑥 𝑛) \infty 𝑛 = 1 \in 𝑋$ be a sequence. Then $𝑎 \in 𝑋$ is a Limit point of $𝑥 𝑛$ iff there exists a convergent subsequence $(𝑥 𝑛 𝑖) \infty 𝑖 = 1 \to 𝑎$ . topology

Proof

proof See @looseAlgebraischeTopologie2010, p. 19

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Limit points are limits of convergent subsequences in a first-countable space

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