How can I show that there exist countable proper elementary extension of structure $(\mathbb{N}, \leq, \cdot, +, 0, 1)$? I'm self-studying model theory and I would appreciate some help in this exercise.
My approach for the proof is as follows:
I will denote the structure $(\mathbb{N}, \leq, \cdot, +, 0, 1)$ as $\mathcal{N}$ with universe $\mathbb{N}$ and language $\mathcal{L} = \{ \leq, \cdot, +, 0, 1 \}$.
Let $\mathcal{L}^* = \mathcal{L} \cup \mathcal{L}_{\mathbb{N}} \cup \{c\}$ where $c$ is a new constant symbol and let $$ T = \mathrm{Diag}_{\mathrm{el}}(\mathcal{N}) \cup \left\{ \underbrace{1 + 1 + \cdots + 1}_{n\text{-times}} < c : \text{for }n = 1, 2, \ldots \right\} $$ If $T_0$ is a finite subset of $T$, then $\mathcal{N} \models T_0$ if we interpret $c$ as large enough natural number. Thus $T$ is finitely satisfable and using Godel's compactness theorem, $T$ is satisfable, so there exist $\mathcal{M} \models T$, so $\mathcal{M} \models \mathrm{Th}(\mathcal{N})$ and $\mathcal{N} \equiv \mathcal{M}$. If $a \in M$ is interpretation of $c$, then $a$ is greater than every natural number, therefore $a \notin \mathbb{N}$, so $\mathbb{N} \subsetneq M$. Using downward Lowenheim–Skolem theorem, there exist elementary substructure $\mathcal{S} \preceq \mathcal{M}$, such that $\mathbb{N} \subsetneq S$ and $$ |\mathcal{N}| \leq |S| + |\mathcal{L}| + \aleph_0 = \aleph_0 + 5 + \aleph_0 = \aleph_0 $$ so $\mathcal{M}$ is countable proper elementary extension of $\mathcal{S}$. Since $\mathcal{S} \models \mathrm{Diag}_{\mathrm{el}}(\mathcal{N})$, then there exist elementary embedding of $\mathcal{N}$ into $\mathcal{S}$, so $\mathcal{N} \preceq \mathcal{S}$, so $\mathcal{M}$ is countable proper elementary extension of $\mathcal{N}$.
However I'm not sure if my proof is correct. I'd be glad of getting any help.