(related to Corollary: Every Distance Is Positive Definite)
Since \(d\) is a distance on \(X\), it follows by the definition of its three properties that \[0=d(x,x)\le d(x,y)+d(y,x)=2d(x,y).\] Consequently, it must be \(0\le d(x,y)\) for all \(x,y\in X\).
