Why can an algebraic extension of a local field be written as a tower of finite extensions?

Question

Let $K$ be a local field and $L$ an algebraic extension of $K$. I found a statement that $L$ can be written as $$ L = \bigcup_{n=0}^\infty L_n, $$ where $L_n$ are finite extensions of $K$ with $L_n \subseteq L_{n+1}$.
I could not find any reference for this but one can show it using the known statement that there are only finitely many extensions of $K$ of a fixed degree.
Does anyone know if it is possible to state a more elementary proof?

Answer 1

$K$ is a completion of a global field $K'$. Every finite extension $E$ of $K$ can be obtained as a completion of a finite extension $E'$ of $K'$. This implies that there are only countably many finite extensions of $K$, as $K'$ and hence $K'[x]$ are countable. Every algebraic extension $L$ of $K$ is the directed union of all finitely generated subextensions $L=\bigcup E_i$. Since there are only countably many, we can choose an enumeration $(E_n)_{n \in \Bbb N}$ and then set $L_n=E_1E_2\dots E_n$.