Field Adjoined Algebraic Elements is Algebraic

Definition
Let $L/K$ be a field extension and $S \subseteq L$ a subset.

If each $x \in S$ is algebraic over $K$ then $K(S)$ is algebraic over $K$.

Proof
Let $S \subseteq L$ be arbitrary, and $x \in K(S)$.

By Field Adjoined Set $x \in K(S)$ if and only if $x \in K(\alpha_1,\ldots,\alpha_n)$ for some $\alpha_1,\ldots,\alpha_n \in S$.

We have that $K(\alpha_1,\ldots,\alpha_n)/K$ is finite by Finitely Generated Algebraic Extension is Finite.

Moreover a Finite Field Extension is Algebraic.

Therefore $x$ is algebraic over $K$.