Let $X$ and $X^{\prime}$ be metric spaces with metrics $d$ and $d^{\prime}$. If $u=\left(x, x^{\prime}\right)$ and $v=\left(y, y^{\prime}\right)$ are any two points of $X \times X^{\prime}$, prove that the formula

$$D(u, v)=\max \left\{d(x, y), d^{\prime}\left(x^{\prime}, y^{\prime}\right)\right\}$$

defines a metric on $X \times X^{\prime}$. If $X=X^{\prime}$, prove that the diagonal $\Delta$ of $X \times X$ is closed in $X \times X$.