Sequential Limits are Limit Points of the Sequence

Last updated Nov 1, 2022

# Definition

Suppose $(X, \tau)$ is a Topological Space

and let

(x_n) \subset X

. Suppose

x_{n} \to x \in X

. Then

x

{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