Complex Modulus Function is Continuous

Theorem
Let $z_0 \in \C$ be a complex number.

Then the complex modulus function is continuous at $z_0$.

Proof
Let $\epsilon>0$.

Let $z \in \C$ be a complex number satisfying $\left\vert{z - z_0}\right\vert < \epsilon$.

By the triangle inequality, $\left\vert{ \left\vert{z}\right\vert - \left\vert{z_0}\right\vert }\right\vert \le \left\vert{z - z_0}\right\vert < \epsilon$.

Hence the result, by the $\epsilon$-$\delta$ definition of continuity (taking $\delta = \epsilon$).