To prove that
If $(\omega, <) \equiv \mathcal{M}$, then there exists a function $f: \omega \to M$ (the domain of $\mathcal{M}$) such that $(\omega, <) \prec_{f} \mathcal{M}$.
where,
the symbol $\equiv$ denotes "elementary equivalence", and $\prec_{f}$ "elementary embedding" with the function $f$.
My thought: (may be totally wrong) I think any function $f$ which is order preserving suffices. However, how to construct such a function or to justify its existence? Furthermore, how to formally prove that $(\omega, <) \prec_{f} \mathcal{M}$ (what about induction on formulas)?
If $\mathcal{M}$ is a model of the theory of $(\omega,<)$ then $\mathcal{M}$ must possess an initial $\omega$ sequence. Because all of these elements are definable in $\mathcal{M}$: The first element, the second etc all first order. Now its easy to check, using the definability of all the elements, that this initial segment is an elementary substructure.