Reverse Triangle Inequality

Theorem
Let $$M = \left({X, d}\right)$$ be a metric space.

Then $$\forall x, y, z \in X: \left|{d \left({x, z}\right) - d \left({y, z}\right)}\right| \le d \left({x, y}\right)$$.

Proof
As $$M = \left({X, d}\right)$$ is a metric space, we have $$\forall x, y, z \in X: d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$$.

So $$d \left({x, y}\right) \ge d \left({x, z}\right) - d \left({y, z}\right)$$ by subtracting $$d \left({y, z}\right)$$ from both sides.

If $$d \left({x, z}\right) - d \left({y, z}\right) \ge 0$$ we are done.

Otherwise, that means $$d \left({x, z}\right) < d \left({y, z}\right)$$, and instead we use $$\forall x, y, z \in X: d \left({y, x}\right) + d \left({x, z}\right) \ge d \left({y, z}\right)$$.

Hence $$d \left({x, y}\right) \ge d \left({y, z}\right) - d \left({x, z}\right)$$.

So combining them both together it follows that $$\forall x, y, z \in X: \left|{d \left({x, z}\right) - d \left({y, z}\right)}\right| \le d \left({x, y}\right)$$.