Complex Plane is Complete Metric Space

Theorem

The complex plane, along with the metric induced by the norm given by the complex modulus, forms a complete metric space.

Proof

Let $z = x + iy$ be a complex number, where $x, y \in \R$.

Now, we can identify the complex number $z$ with the ordered pair $\left( x, y \right) \in \R^2$.

The norm on $\C$ given by the complex modulus is then identical to the Euclidean norm on $\R^2$.

Therefore, metric on $\C$ induced by the norm given by the complex modulus is identical to the Euclidean metric on $\R^2$ induced by the Euclidean norm.

By Euclidean Space is Complete Metric Space, $\C$ is a complete metric space.

$\blacksquare$

Sources

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• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $3.11c$