Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $X\times Y$ be the Cartesian product of the underlying sets, and let $d_{X\times Y}$ be the product metric, which is a metric by claim 1 of The Product Metric is a Metric. Let $\mathbb{N}$ denote the natural numbers. Write $\mathcal{T}_X$, $\mathcal{T}_Y$ and $\mathcal{T}_{X\times Y}$ for the collections of subsets open in $(X,d_X)$, in $(Y,d_Y)$ and in $(X\times Y,d_{X\times Y})$ respectively; each is a topology on the corresponding set by The Open Subsets of a Metric Space Form a Topology.

Then the following hold.

1. (Coordinatewise convergence.) Let $(z_m)_{m\in\mathbb{N}}$ be a sequence in $X\times Y$, written $z_m=(x_m,y_m)$ with $x_m\in X$ and $y_m\in Y$, and let $z=(x,y)\in X\times Y$. Then $(z_m)_{m\in\mathbb{N}}$ converges to $z$ in $(X\times Y,d_{X\times Y})$ if and only if $(x_m)_{m\in\mathbb{N}}$ converges to $x$ in $(X,d_X)$ and $(y_m)_{m\in\mathbb{N}}$ converges to $y$ in $(Y,d_Y)$.

2. (Products of sequentially compact sets.) Let $K\subseteq X$ be sequentially compact in $(X,d_X)$ and let $M\subseteq Y$ be sequentially compact in $(Y,d_Y)$. Then $K\times M$ is sequentially compact in $(X\times Y,d_{X\times Y})$.

3. (Products of dense sets.) Let $D\subseteq X$ be dense in $X$ for $\mathcal{T}_X$ and let $E\subseteq Y$ be dense in $Y$ for $\mathcal{T}_Y$. Then $D\times E$ is dense in $X\times Y$ for $\mathcal{T}_{X\times Y}$. If moreover $D$ and $E$ are countable, then $D\times E$ is countable.