Real Number is Limit Point of Rational Numbers in Real Numbers

Theorem

Let $\R$ be the set of real numbers.

Let $\Q$ be the set of rational numbers.

Let $x \in \R$.

Then $x$ is a limit point of $\Q$.

Hence the interesting case of a $\Q$ is countable set $\Q$ whose set of limit points is uncountable.

Proof

Let $\epsilon \in \R_{>0}$ be arbitrary.

Consider the open interval $\openint x {x + \epsilon}$.

Then by Rationals are Everywhere Dense in Topological Space of Reals, there exists a rational number in $\openint x {x + \epsilon}$.

Hence the result by definition of limit point.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.10 \ \text {(c)}$