Rational vs real metric space

Question

How to prevent, in a lesson that deals with basic mathematics, that we give two definitions of a metric ? Because there is one, which takes value in $\mathbb{Q}$, to build $\mathbb{R}$, that we do not know already. And one that conveniently takes value in $\mathbb{R}$, for other cases.

Answer 1

Given the comments below my question, here are some leads :

1. Define $\mathbb{R}$ as classes of Cauchy sequences (defined as the sequences which verify : $\forall\epsilon\in\mathbb{Q},\epsilon>0\Rightarrow(\exists N\in\mathbb{N},\forall n\geq N,\forall p\geq0,\lvert u_{n+p}-u_{n}\rvert_\mathbb{Q}\leq\epsilon)$, with $\lvert.\rvert_\mathbb{Q}$ being the absolute value in $\mathbb{Q}$). Then there is a strictly increasing ring morphism $i:\mathbb{Q}\rightarrow\mathbb{R}$.
   
   Define a metric (as usual, with value in the constructed $\mathbb{R}$). Remark that $(x,y)\overset{d}{\rightarrow} \lvert i(x)-i(y)\rvert_\mathbb{R}$ is a metric on $\mathbb{Q}$ (rigourously, $\lvert.\rvert_\mathbb{Q}$ does not have value in $\mathbb{R}$).
   
   Define the completion of a metric space. The Cauchy sequences defined above are exactly the Cauchy sequences for this metric. And we have $\lvert x-y\rvert_\mathbb{R}=\lim_{n\to\infty}\lvert i(x_n)-i(y_n)\rvert$ with $(x_n)$ and $(y_n)$ being representatives of $x$ and $y$. So actually, $\mathbb{R}$ is the completion of $\mathbb{Q}$.
2. Use Dedekind cuts.

It is interesting to note that to define a metric with value in $\mathbb{R}$ or with value in $\mathbb{R'}$ which are both ordered field which are archimedian and complete lead to the same concept, as there is a strictly increasing ring isomorphism $R$ between $\mathbb{R}$ and $\mathbb{R'}$. So the metrics with value in $\mathbb{R}$ and those with value in $\mathbb{R'}$ are in bijection : $d\to R\circ d$.
$d$ and $R\circ d$ are topologically equivalent and lead to the same Cauchy sequences, isometries, uniform continuity and Lipschitz functions.