Open Unit Interval on Rational Number Space is Bounded but not Compact

Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.

Then:
 * $\openint 0 1 \cap \Q$ is totally bounded but not compact

where $\openint 0 1$ is the open unit interval.

Proof
We first show that $\openint 0 1 \cap \Q$ is totally bounded.

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

By the Archimedean Property of $\R$:
 * $\exists n \in \N: \dfrac 1 n < \epsilon$

We pick the numbers $\dfrac i n \in \openint 0 1 \cap \Q$, where $i \in \N$ and $0 \le i \le n$.

Then for all $x \in \openint 0 1 \cap \Q$:

This shows that $\openint 0 1 \cap \Q$ totally bounded.

From the Heine-Borel Theorem on a metric space, $\openint 0 1 \cap \Q$ is compact it is both totally bounded and complete.

From Rational Number Space is not Complete Metric Space it follows that $\openint 0 1 \cap \Q$ is not compact.