Chebyshev Distance on Real Vector Space is Metric/Proof 1

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Theorem

The Chebyshev distance on $\R^n$:

$\ds \forall x, y \in \R^n: \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n {\size {x_i - y_i} }$

is a metric.


Proof

This is an instance of the Chebyshev distance on the cartesian product of metric spaces $A_1, A_2, \ldots, A_3$.

This is proved in Chebyshev Distance is Metric.

$\blacksquare$


Sources