Proof of Continuity of the Projections and of the Distance Function on a Product Metric Space

For real numbers $s,t$ we write $s<t$ to mean that $s\le t$ and $s\ne t$, and $s-t$ for $s+(-t)$; we use the order arithmetic of Elementary Order Arithmetic in an Ordered Field and the properties of the absolute value recorded in Properties of the Absolute Value in an Ordered Field. We also use that if $a\le b$ and $c\le e$ then $a+c\le b+e$, which follows from claims 2 and 3 of Elementary Arithmetic in an Ordered Field with the addition axioms of the underlying field.

Claim 1. Let $p=(x,y)\in S$ and let $\varepsilon$ be a real number with $0<\varepsilon$. Put $\delta=\varepsilon$. Let $q=(x',y')\in S$ satisfy $d_{X\times Y}(p,q)<\delta$. By claim 3 of The Product Metric is a Metric this gives $d_X(x,x')<\varepsilon$, and condition 3 in the definition of a metric gives

$$d_X\bigl(\pi_X(q),\pi_X(p)\bigr)=d_X(x',x)=d_X(x,x')<\varepsilon .$$

Thus $\pi_X$ is continuous at $p$ relative to $S$, and since $p\in S$ was arbitrary it is continuous on $S$. The argument for $\pi_Y$ is the same, using the inequality for $d_Y$ in claim 3 of The Product Metric is a Metric.

Claim 2. Let $p=(x,y)\in T$ and let $\varepsilon$ be a real number with $0<\varepsilon$. Set $2=1+1$ and $\delta=\varepsilon\cdot 2^{-1}$, so that $0<\delta$ and $\delta+\delta=\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. Let $q=(x',y')\in T$ satisfy $d_{Z\times Z}(p,q)<\delta$. By claim 3 of The Product Metric is a Metric we have $d_Z(x,x')<\delta$ and $d_Z(y,y')<\delta$.

Two applications of condition 4 in the definition of a metric, together with condition 3, give

$$d_Z(x',y')\le d_Z(x',x)+d_Z(x,y')\le d_Z(x,x')+d_Z(x,y)+d_Z(y,y') ,$$

and hence, by claim 3 of Elementary Arithmetic in an Ordered Field and the addition axioms of the field,

$$d_Z(x',y')-d_Z(x,y)\le d_Z(x,x')+d_Z(y,y') .$$

By claim 3 of Elementary Order Arithmetic in an Ordered Field, from $d_Z(x,x')<\delta$ and $d_Z(y,y')\le\delta$ we get $d_Z(x,x')+d_Z(y,y')<\delta+\delta=\varepsilon$, so claim 2 of that lemma gives

$$d_Z(x',y')-d_Z(x,y)<\varepsilon .$$

Interchanging the roles of $p$ and $q$ in the same argument gives $d_Z(x,y)-d_Z(x',y')<\varepsilon$, that is $-\varepsilon<d_Z(x',y')-d_Z(x,y)$ by claim 4 of Elementary Order Arithmetic in an Ordered Field and the field identity $-(a-b)=b-a$.

By claim 9 of Properties of the Absolute Value in an Ordered Field the two displayed strict bounds give

$$d_{\mathbb{R}}\bigl(\rho(q),\rho(p)\bigr)=\bigl|d_Z(x',y')-d_Z(x,y)\bigr|<\varepsilon ,$$

using the description of $d_{\mathbb{R}}$ in The Absolute Value Metric on the Real Line. Thus $\rho$ is continuous at $p$ relative to $T$, and since $p\in T$ was arbitrary it is continuous on $T$.