Consider language $\mathcal{L}$ consisting of binary relation symbol <. Let $\mathfrak{A}_{0}$ be $\mathcal{L}$-structure with universe $A_{0}=\{0,1,\dots\}$. Let $\mathfrak{A}_{n}$ be the $\mathcal{L}$-structure with universe $A_{n} := \{-n,\dots,-1,0,1,2,\dots\}$ with natural interpretation of <.
Show that Th$(\mathfrak{A}_{n})$=Th$(\mathfrak{A}_{i})$ for all $i \in \mathbb{N}$.
I am not sure how to show that formally. We have to show the elementary equivalence. The theory is the set of $\mathcal{L}$-sentences, so formulas without free variables. The only difference between the structures is the universe. So all the structures satisfy the same sentences, since sentences don't include any free variables and the relation symbol is always the same.
Am I missing something here or how could I write this up more formally?