Suppose I have $\preceq$, a total order on $\mathbb R^n$. I wish to show that there is a utility function $u:\mathbb R^n\to\mathbb R$ such that $x\preceq y \leftrightarrow u(x)\leq u(y)$.
I came up with a constructive proof, which might be best explained with an example:
Suppose we have that $x_1\preceq x_2$. We can assign $x_1$ utility 0 and $x_2$ utility 1. If $x_3$ is smaller than $x_1$ it gets utility -1, if it's bigger than $x_2$ it gets utility 2, and if it's between it gets utility $1/2$. Continue indefinitely.
Is this a valid proof? My concern is that I might be assuming that $\mathbb R^n$ is recursively enumerable.