Define $d(x,y) = |x - y|^2$ . Explain why d is not a metric.
I have seen that the first two properties of a metric hold, so naturally it's left to prove that the triangle inequality does not hold. But after some algebra, I get stuck at the following:
$2|y|(|x| + |z|)-2|x||z|- 2|y|^2<= 0 $
Am I missing something obvious here? I am not sure how to prove that this inequality doesn't hold from here.