Proof of The Euclidean Distance on the Real Line is the Absolute Value Metric

By the definition of the Euclidean distance in the case $n=1$, $d_{E}(s,t)$ is the nonnegative real number whose square is $(s-t)^{2}$.

The number $|s-t|$ has both properties: it is nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field, and its square is $(s-t)^{2}$ by claim 4 of that lemma, applied with both arguments equal to $s-t$.

Distinct nonnegative real numbers have distinct squares. Indeed, suppose $0\le\alpha$, $0\le\beta$ and $\alpha<\beta$. Then $0<\beta$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, so $\beta\alpha<\beta\beta$ by claim 10 of that lemma; and $\alpha\alpha\le\beta\alpha$, this being an equality when $\alpha=0$ and following from claim 10 with multiplier $\alpha$ when $0<\alpha$. Claim 2 then gives $\alpha^{2}<\beta^{2}$, so $\alpha^{2}\ne\beta^{2}$.

Hence the nonnegative real number with square $(s-t)^{2}$ is unique, and

$$d_{E}(s,t)=|s-t|=d_{\mathbb{R}}(s,t)$$

by The Absolute Value Metric on the Real Line.

Since $s$ and $t$ were arbitrary, $d_{E}$ and $d_{\mathbb{R}}$ are the same function on $\mathbb{R}\times\mathbb{R}$. Every notion in the remaining assertions is defined purely in terms of that function: the open subsets of a metric space are defined from its metric, the topology of Metric Open Sets Form a Topology is the collection of those open subsets, and compactness of a subset is defined from that topology. Equal metrics therefore give literally equal collections of open sets, equal topologies, and the same compact subsets.