Show that if U is a principal ultrafilter then the canonical inmersion is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$

Question

Show that: If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$

Answer 1

Anyone unfamiliar with the context of this question should read about it here. The canonical embedding $j$ sends an element of the model $a \in |\mathcal A|$ to the equivalence class of the constant sequence $\{a,a,a,\ldots\}$ under the ultrafilter $U$.

Because of Los' Theorem, the canonical embedding $j$ is always injective (if $\mathcal A \models a \ne b$, then $Ult_U(\mathcal A)\models j(a) \ne j(b)$). Hence we only need to show that $j$ is surjective. If an ultrafilter $U$ in $\kappa$ is principal, then it takes the form $\{X \subset \kappa: \beta \in X\}$ for some $\beta < \kappa$. Therefore, if $b = \{b_\alpha:\alpha<\kappa\} \subset \mathcal A^\kappa$ and $[b]_U \in Ult_U(\mathcal A)$, then $j(b_\beta)=[b]_U$ by Los' Theorem.