Let DLO be the theory of dense linear ordering with no endpoints. I'm trying to show:
Let $K$ be a countable subset of R which models DLO. Prove that $(K,<)$ is an elementary submodel of $(\mathbb{R},<)$.
I wanted to show this using the Tarski-Vaught Criterion so, I was trying to show that for any formula of the theory on $\mathbb{R}$ with a variable $b\in (\mathbb{R},<)$ I can find $c\in K$ for which the formula holds. However this proof seems to require an injective map from $\mathbb{R}$ to $K$ which seems impossible because that would imply that $\omega$ is less than the continuum.
Let $a_1,\ldots, a_n\in K$ and suppose there exists $b\in\Bbb R$ with $\phi(b,a_1,\ldots, a_n)$. We want to find $c\in K$ with $\phi(c,a_1,\ldots, a_n)$. If $b\in K$, we are done, so we assume $b\notin K$ and in particular $b\ne a_i$ for all $i$. Then $b$ is in one of the (at most) $n+1$ open intervals of $\Bbb R$ determined by the $a_i$. Each such interval contains elements of $K$ because (for the outer, unbounded intervals) $(K,{<})$ has no endpoints and (for the inner, bounded intervals) $(K,{<})$ is dense. Pick $c\in K$ from the same interval as $b$. Now prove the desired claim by structural induction on $\phi$ ...