Composite of Continuous Mappings between Metric Spaces is Continuous/Proof 2
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Theorem
Let $M_1 = \struct {X_1, d_1}$, $M_2 = \struct {X_2, d_2}$ and $M_3 = \struct {X_3, d_3}$ 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 $\map f a \in X_2$ 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
$\blacksquare$