Identity Mapping is Continuous/Metric Space

Theorem
Let $M = \left({X, d}\right)$ be a metric space.

The identity mapping $I_X: X \to X$ defined as:
 * $\forall x \in X: I_X \left({x}\right) = x$

is a continuous mapping.

Proof
Let $a \in X$.

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

Let $\delta = \epsilon$.

Then: