I cannot imagine that this application of the triangle inequality Prove $| |y-x| - |z-y| | \leq |x-z|$ hasn't been asked and answered before, but I could not find any duplicate.

@MartinR Isn't this basically reverse triangle inequality?

If we have $||a|-|b|| \le |a-b|$ and we substitute: $a=y-x$ and $b=y-z$.

I see that this is actually in one answer. :-)

Yes, with that substitution. But I think that I found a suitable candidate now, by interpreting |x-y| as d(x,y) for a metric.