Pigeonhole distance bound from a partition ($m+1\text{ points in }m\text{ sets of diameter }\le D⟹d(x_i,x_j)\le D$)

Suppose a subset of a metric space is covered by $m$ sets, each of diameter at most $D$. Assign each point to one covering set containing it, choosing a single set at shared boundaries. Among $m+1$ points, the pigeonhole principle puts two in a common set, and their distance is at most $D$. The pieces need not be equal or congruent. This freedom permits better distance bounds than dividing a region only into equal rectangles.