Sequential Limits are Limit Points of the Sequence

Last updated Nov 1, 2022

Suppose $(X, \tau)$ is a Topological Space and let $(x_n) \subset X$. Suppose $x_{n} \to x \in X$. Then $x$ is a Limit Point of ${x_{n} : n \in \mathbb{N}}$.

Suppose $U \subset X$ is Open. Then, by definition of Sequence Convergence, there exists $N \in \mathbb{N}$ so that $\forall n \geq N$, $x_{n} \in U$. Thus, $U \cap {x_{n} : n \in \mathbb{N}} \supset {x_{n} : n \geq N}$ and $x$ is a Limit Point of our Sequence.