If $X=\mathbb{R}^2$. For $x=(x_1,x_2)$, $y=(y_1,y_2)$ define
$d_{1/2}(x,y)=\left(|x_1-y_1|^{1/2}+|x_2-y_2|^{1/2}\right)^2\;.$
Prove or disprove $(X,d_{1/2})$ is a metric space.
Attempt at a solution: After multiplying it out, it seems to boil down to $2|x_1-y_1|^{1/2}|x_2-y_2|^{1/2}$ satisfying the triangle inequality. However, I can't seems to prove or disprove this one way or another.
Intution, however, is leading me towards the fact that it is in fact not a metric space.