Omega-Accumulation Point of Underlying Set of Sequence of Reciprocals and Reciprocals + 1

Theorem
Let $\sequence {a_n}$ denote the sequence defined as:

Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology.

Let $S$ denote the set of terms of $\sequence {a_n}$ considered as a subset of $\struct {\R, \tau_d}$.

Then $0$ is an $\omega$-accumulation point of $S$.

Proof
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

Then the open interval $\openint {-\epsilon} \epsilon$ contains $0$ and all elements $a_m$ of $S$ such that $0 < \dfrac 1 m < \epsilon$.

We have that:
 * $\forall n \in \N: n \ge m$

have the property that $0 < \dfrac 1 n < \epsilon$.

Hence there are a countably infinite number of terms of $\sequence {a_n}$ such that $a_n \in \openint {-\epsilon} \epsilon$.

Hence the result by definition of $\omega$-accumulation point of $S$.