[[First countability axiom]]
# Limit points are limits of convergent subsequences in a first-countable space

Let $X$ be a [[First countability axiom|first-countable]] topological space and $(x_n)_{n=1}^\infty \in X$ be a sequence.
Then $a \in X$ is a [[Limit point]] of $x_{n}$ iff there exists a [[Convergence|convergent]] [[subsequence]] $(x_{n_{i}})_{i=1}^\infty \to a$. #m/thm/topology 

> [!missing]- Proof
> #missing/proof 
> See [[@looseAlgebraischeTopologie2010]], p. 19

#
---
#state/develop | #lang/en | #SemBr