The Product Metric is a Metric

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $d_{X\times Y}$ be the product metric on $X\times Y$. Let $\mathbb{R}$ be the set of real numbers with the addition and the order $\le$ of its ordered field structure, and for $s,t\in\mathbb{R}$ write $s<t$ to mean that $s\le t$ and $s\ne t$. Let $p=(x_1,y_1)$ and $q=(x_2,y_2)$ be points of $X\times Y$ and let $r\in\mathbb{R}$. Then the following hold.

1. (Metric) $d_{X\times Y}$ is a metric on $X\times Y$, so that $(X\times Y,d_{X\times Y})$ is a metric space.

2. (Domination of the factors) $d_X(x_1,x_2)\le d_{X\times Y}(p,q)$ and $d_Y(y_1,y_2)\le d_{X\times Y}(p,q)$.

3. (Strict bounds) $d_{X\times Y}(p,q)<r$ if and only if both $d_X(x_1,x_2)<r$ and $d_Y(y_1,y_2)<r$.