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

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 $\map f a \in X_2$ for all $\map f a \in X_2$.

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