Composite of Continuous Mappings between Metric Spaces is Continuous/Proof 1

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 $f: M_1 \to M_2$ be continuous at $a \in X_1$.

Let $g: M_2 \to M_3$ be continuous at $f \left({a}\right) \in X_2$.

Then their composite $g \circ f: M_1 \to M_3$ is continuous at $a \in X_1$.

Proof
From Metric Induces Topology, the metric spaces described are topological spaces.

The result follows from Composite of Continuous Mappings is Continuous.