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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