On $\displaystyle X=\mathbb R^2$ consider $\displaystyle d(x,y)={{\left( {{\left| {{x}_{1}}-{{y}_{1}} \right|}^{\frac{1}{2}}}+{{\left| {{x}_{2}}-{{y}_{2}} \right|}^{\frac{1}{2}}} \right)}^{2}},$ where $\displaystyle x=(x_1,x_2)$ and $\displaystyle y=(y_1,y_2).$ Is it $\displaystyle (X,d)$ a metric space?

I think is not, because of the triangle inequality.

What do you guys think?