[proofplan] We use the diagonal embedding of $M$ into its ultrapower. The ultrapower elementary embedding theorem says that this diagonal map is elementary. Since elementary maps preserve and reflect truth of all formulas with parameters, applying this to formulas with no free variables shows that $M$ and $M^I/\mathcal U$ satisfy exactly the same $L$-sentences. [/proofplan] [step:Define the diagonal map into the ultrapower] For each $a \in M$, let \begin{align*} c_a: I &\to M \\ i &\mapsto a \end{align*} denote the constant function with value $a$. Define the diagonal map \begin{align*} d: M &\to M^I/\mathcal U \\ a &\mapsto [c_a]_{\mathcal U}. \end{align*} Here $[c_a]_{\mathcal U}$ denotes the $\mathcal U$-equivalence class of $c_a$ in the [ultrapower](/page/Ultrapower) by the [ultrafilter](/page/Ultrafilter) $\mathcal U$. The hypotheses required by the [Ultrapower Elementary Embedding Theorem](/theorems/???) are exactly the hypotheses in force: $M$ is an $L$-structure, $I$ is non-empty, and $\mathcal U$ is an ultrafilter on $I$. Therefore the theorem applies to the diagonal map $d$ and gives that $d$ is an [elementary embedding](/page/Elementary%20Embedding) of $L$-structures. [/step] [step:Apply elementarity to sentences] Let $\sigma$ be an arbitrary $L$-sentence. Since $\sigma$ has no free variables, elementarity of \begin{align*} d: M \to M^I/\mathcal U \end{align*} gives \begin{align*} M \models \sigma \iff M^I/\mathcal U \models \sigma. \end{align*} Indeed, this is the parameter-free case of the defining property of an elementary embedding: for every $L$-formula $\varphi(x_1,\dots,x_n)$ and every tuple $(a_1,\dots,a_n) \in M^n$, \begin{align*} M \models \varphi(a_1,\dots,a_n) \iff M^I/\mathcal U \models \varphi(d(a_1),\dots,d(a_n)). \end{align*} Taking $n = 0$ and $\varphi = \sigma$ gives the displayed equivalence. [/step] [step:Conclude elementary equivalence] Since $\sigma$ was an arbitrary $L$-sentence, $M$ and $M^I/\mathcal U$ satisfy exactly the same $L$-sentences. By the definition of [elementary equivalence](/page/Elementary%20Equivalence), \begin{align*} M^I/\mathcal U \equiv M. \end{align*} This proves the theorem. [/step]