Identity Mapping is Continuous/Metric Space

Theorem
Let $M = \struct {A, d}$ be a metric space.

The identity mapping $I_A: A \to A$ defined as:
 * $\forall x \in A: \map {I_A} x = x$

is a continuous mapping.

Proof
Let $a \in A$.

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

Let $\delta = \epsilon$.

Then: