Cantor Space is Complete Metric Space

Theorem
Let $\left({\mathcal C, \tau_d}\right)$ be the Cantor set considered as a topological subspace of the real number space $\R$ under the Euclidean topology $\tau_d$.

Then $\mathcal C$ is a complete metric space.

Proof
We have that the Cantor set is a metric subspace of the real number space $\R$, and hence a metric space.

We also have Cantor Set Compact in Real Number Space.

The result follows from Compact Metric Space is Complete.