The Euclidean metric on is the standard metric on this space. The distance between two elements and is given by . It is straight-forward to show that this is symmetric, non-negative, and 0 if and only if . Showing that the triangle inequality holds true is somewhat more difficult, although it should be intuitively clear because it is properties of the Euclidean metric which motivate the definition of a metric.
Proof of the triangle inequality
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