Irrational Number Space is Complete Metric Space

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

Then $\struct {\R \setminus \Q, \tau_d}$ is a complete metric space.

Proof
Let $d: \R \times \R \to \R$ be the Euclidean plus metric:
 * $\ds \map d {x, y} := \size {x - y} + \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$

Let $\sequence {x_n}$ converge in $\struct {\R \setminus \Q, \tau_d}$ to a rational number.

For each $x_i$ in $\sequence {x_n}$, there exists $j > i$ such that $\map d {x_i, x_j} \ge \size {x_i - x_j} + 2^{-k}$ for some $k$.

Hence $\sequence {x_n}$ is not a Cauchy sequence.

Thus if $\sequence {x_n}$ is a Cauchy sequence it converges to an irrational number.