Limit Point of Sequence is Adherent Point of Range
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Theorem
Let $T = \struct{S, \tau}$ be a topological space.
Let $\sequence{x_n}$ be a sequence in $S$.
Let $\alpha$ be a limit of $\sequence{x_n}$.
Then $\alpha$ is an adherent point of $\set{x_n: n \in \N}$.
Proof
By definition of Limit of sequence:
- $\forall U \in \tau : \exists N \in \N : \forall n \ge N : x_n \in U$
Hence:
- $\forall U \in \tau : U \cap \set{x_n: n \in \N} \ne \O$.
By definition $\alpha$ is an adherent point of $\set{x_n: n \in \N}$.
$\blacksquare$