Set of Reciprocals of Positive Integers is Nowhere Dense in Reals

Theorem

Let $N$ be the set defined as:

$N := \set {\dfrac 1 n: n \in \Z_{>0} }$

where $\Z_{>0}$ is the set of (strictly) positive integers.

Let $\R$ denote the real number line with the usual (Euclidean) metric.

Then $N$ is nowhere dense in $\R$.

Proof

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

Hence:

$\map \cl N = \set {\dfrac 1 n: n \in \Z_{>0} } \cup \set 0$

where $\map \cl N$ denotes the closure of $N$ in $\R$.

Trivially, $\map \cl N$ contains no open real intervals.

Hence no subset of $\map \cl N$ is open in $\R$.

Hence the union of all the subset of $\map \cl N$ which are open in $\R$ is empty.

That is, by definition, the interior of $N$ is empty.

That is:

$\paren {\map \cl N}^\circ = \O$

and the result follows by definition of nowhere dense.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Example $3.7.27$