Suppose that $\left(X, d_{X}\right)$ and $\left(Y, d_{Y}\right)$ are metric spaces. Show that the definition

$$d\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=d_{X}\left(x_{1}, x_{2}\right)+d_{Y}\left(y_{1}, y_{2}\right)$$

defines a metric on the product $X \times Y$, under which the projection map $\pi: X \times Y \rightarrow Y$ is continuous.

If $\left(X, d_{X}\right)$ is compact, show that every sequence in $X$ has a subsequence converging to a point of $X$. Deduce that the projection map $\pi$ then has the property that, for any closed subset $F \subset X \times Y$, the image $\pi(F)$ is closed in $Y$. Give an example to show that this fails if $\left(X, d_{X}\right)$ is not assumed compact.