Suppose $(X_{1}, d_{1}), (X_{2}, d_{2})$ are two metric space, and $X = X_{1} \times X_{2}$, then for $\boldsymbol{x}, \boldsymbol{y} \in X, \boldsymbol{x} = (x_{1}, x_{2}), \boldsymbol{y} = (y_{1}, y_{2})$ where $x_{1}, y_{1} \in X_{1}$ and $x_{2}, y_{2} \in X_{2}$, prove
$d(\boldsymbol{x}, \boldsymbol{y}) = \sqrt{(d_{1}(x_{1}, y_{1}))^{2} + (d_{2}(x_{2}, y_{2}))^{2}} $ is a metric