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Bounded

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Intuitively, a set is bounded if the distances between its points are all less than some finite real number (the bound). Formally, we say that a subset S of a metric space (X, d) (such as the standard Euclidean plane, \mathbb{R}^2 with distance d((x, y), (w, z)) = \sqrt{(x - w)^2 + (y - z)^2}), is bounded if for some x \in X there exists some M \in \mathbb{R}_{\geq 0} such that for all s \in S, d(s, x) < M.

Note that if a set S is bounded, the choice of x is immaterial if are willing to change the bound: we have by the triangle inequality that d(y, s) \leq d(y, x) + d(x, s) \leq d(x, y) + M for all s \in S.

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