The complete problem is as follows :
Let $V = \{ +, \cdot , 0, 1\}$ and let $\mathcal{N} = \{ \mathbb{N} , + , \cdot , 0, 1\}$. Also consider the set of all $V$-sentences which are true in $\mathcal{N}$, that is $T = Th(\mathcal{N})$. Let ${\phi}(\mathcal{M})$ denote the $V$-definable subset of a structure $\mathcal{M}$. Suppose that ${\phi}(x)$ is a $V$-formula and ${\phi}(\mathcal{N})$ is finite. Show that if $\mathcal{M} \equiv \mathcal {N}$ then $\mathcal{M} \succsim \mathcal{N}$.
There is a hint in this exercise saying that for every $n\in \mathbb{N}$ there is term without variables such that $(t_n)^{\mathcal{N}} = n$ and that we can consider the function $n \mapsto (t_n)^{\mathcal{M}}$.
My thoughts was about starting from the fact that $\mathcal{M} \equiv \mathcal{N}$. That is true if and only if $Th(\mathcal{M}) = Th(\mathcal{N})$. However I am stuck and I do not seem to understand how to continue.
Any help would be very much appreciated.