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Euclidean metric

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The Euclidean metric on \mathbb{R}^n is the standard metric on this space. The distance d(\mathbf{x, y}) between two elements \mathbf{x} = (x_1, x_2, \ldots, x_n) and \mathbf{y} = (y_1, y_2, \ldots, y_n) is given by d(\mathbf{x, y}) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \ldots + (x_n - y_n)^2}. It is straight-forward to show that this is symmetric, non-negative, and 0 if and only if \mathbf{x = y}. 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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