Distance-Preserving Mapping is Injection of Metric Spaces
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Theorem
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $\phi: M_1 \to M_2$ be a distance-preserving mapping.
Then $\phi$ is an injection.
Proof
Let $a, b \in A_1$ and suppose that $\map \phi a = \map \phi b$.
Then by the definition of a metric space:
- $\map {d_2} {\map \phi a, \map \phi b} = 0$
By the definition of a distance-preserving mapping then:
- $\map {d_1} {a, b} = 0$
Thus by the definition of a metric space:
- $a = b$
Hence $\phi$ is injective.
$\blacksquare$