Closure of Integer Reciprocal Space

Theorem
Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
 * $A := \left\{{\dfrac 1 n : n \in \Z_{>0}}\right\}$

Let $\left({A, \tau_d}\right)$ be the integer reciprocal space under the usual (Euclidean) topology.

Then:
 * $A^- = A \cup \left\{{0}\right\}$

where $A^-$ denotes the closure of $A$ in $\R$.

Proof
By definition, the closure of $A$ is:
 * $A \cup A'$

where $A'$ is the derived set of $A$.

By definition of derived set, $A'$ consists of all the limit points of $A$ in $\R$.

From Zero is Limit Point of Integer Reciprocal Space, the only limit point of $A$ is $0$.

Hence the result:
 * $A^- = A \cup \left\{{0}\right\}$