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 of a metric space (such as the standard Euclidean plane, with distance ), is bounded if for some there exists some such that for all , .
Note that if a set is bounded, the choice of is immaterial if are willing to change the bound: we have by the triangle inequality that for all .
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