Order preserving map from interval $(0,1)$ to the ultrafilter

Question

Let $\mathcal{A}= \prod_{n \in \mathbb{N}} \mathcal{A}_n /\mathcal{U}$, where $\mathcal{A}_n=(\{0, 1, \dots, n\},<)$ and $\mathcal{U}$ is a non-principal ultrafilter of $\mathbb{N}$.

Does there exist a function $f:(0,1) \to A$ such that $f(x)<^{\mathcal{A}}f(y)$ whenever $x<y$?

I see that $f$ won't be surjective, since $\mathcal{A}$ contains a least element $[0,0, \dots]$ but $(0,1)$ has no least element.

Answer 1

For a given $r\in(0,1)$ choose $(i_n)_{n\in\omega}$ be such that $\lim_{n\to\infty}i_n/n=r$.

Then $f(r)=[(i_n)]_{\mathcal U}$,