Composite of Continuous Mappings between Metric Spaces is Continuous/Proof 2

Theorem
Let $M_1 = \left({X_1, d_1}\right), M_2 = \left({X_2, d_2}\right), M_3 = \left({X_3, d_3}\right)$ be metric spaces.

Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be continuous mappings.

Then their composite $g \circ f: M_1 \to M_3$ is continuous.

Proof
Let $f$ and $g$ be continuous mappings.

By definition:
 * $f$ is continuous at $a \in X_1$ for all $a \in X_1$
 * $g$ is continuous at $f \left({a}\right) \in X_1$ for all $f \left({a}\right) \in X_1$.

The result follows from Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point