I want to prove that ((0,1), <) is an elementary substructure of $(\mathbb{R}, <)$ i.e. $((0,1), <) \preceq (\mathbb{R}, <)$
The first hint is to prove that there exists an automorphism $h$ of $(\mathbb{R}, <)$ such that $h(a_j)=b_j$ for $a_1 < a_2 < \cdot \cdot \cdot <a_n$ and $b_1 < b_2 < \cdot \cdot \cdot <b_n$.
I'm not sure how I can construct such automorphism. Unlike field automorphism I can't come up with any useful properties from the definition of isomorphism in the First Order Logic.
The second hint is Tarski-Vaught test.
Any hint or solution please?