Let $\varphi$ be an $L$ -sentence, and let $\mathcal{A}$ be the unique model of S of cardinality $\kappa$ . Suppose $\mathcal{A}\vDash\varphi$ . Then if $\mathcal{B}$ is any model of $S$ then by the upward and downward Lowenheim-Skolem theorems, there is a model $\mathcal{C}$ of $S$ which is elementarily equivalent to $\mathcal{B}$ such that $|\mathcal{C}|=\kappa$ . Then $\mathcal{C}$ is isomorphic to $\mathcal{A}$ , and so $\mathcal{C}\vDash\varphi$ , and $\mathcal{B}\vDash\varphi$ . So $\mathcal{B}\vDash\varphi$ for all models $\mathcal{B}$ of $S$ , so $S\vDash\varphi$ .

Similarly, if $\mathcal{A}\vDash\lnot\varphi$ then $S\vDash\lnot\varphi$ . So $S$ is complete.$\square$