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

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $\left \langle {x_n} \right \rangle$ be a sequence of distinct terms of $S$.

Let $\alpha$ be an accumulation point of $\left \langle {x_n} \right \rangle$.


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


Proof

Let $U$ be an open set of $T$ containing $\alpha$.

By definition of accumulation point of $\left \langle {x_n} \right \rangle$, $U$ contains infinitely many terms of $\left \langle {x_n} \right \rangle$.

As $\left \langle {x_n} \right \rangle$ is a sequence of distinct terms: $U$ contains infinitely many elements of $\left\{ {x_n: n \in \N}\right\}$.

Thus by definition $\alpha$ is an $\omega$-accumulation point of $\left\{ {x_n: n \in \N}\right\}$.

$\blacksquare$


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