Suppose we have $A$ and $B$ as subsets of a metric space $(E, d)$. Is it true that for any subset $C$ of $E$, $d(A, B) \leq d(A, C) + d(C, B)$?
2026-02-23 11:44:43.1771847083
General Triangular Inequality for distance between subsets.
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As a counterexample:
let $E =\mathbb{R}$.
let $A = ]-\infty, 1]$ and $B = [3, \infty[$. Take $C = [0, 4]$. It is evident that $d(A, C) + d(C, B) = 0 < d(A, B) = 2$