Limit Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $\sequence {x_n}$ be a sequence of distinct terms of $S$.

Let $\alpha$ be a limit point of $\sequence {x_n}$.

Then $\alpha$ is also an $\omega$-accumulation point of $\set {x_n: n \in \N}$.

Proof

Let $\alpha$ be an limit point of $\sequence {x_n}$.

From Limit of Sequence is Accumulation Point‎, $\alpha$ is an accumulation point of $\sequence {x_n}$.

From Accumulation Point of Sequence of Distinct Terms is Omega-Accumulation Point of Range, $\alpha$ is an $\omega$-accumulation point of $\set {x_n: n \in \N}$.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Limit Points