My homework problem is:
If he have fields $k\subseteq L\subseteq K$ such that $K/k$ is normal then $K/L$ is normal for any intermediate field $L$.
The case for a finite extension is almost trivial since a finite normal extension $K/k$ is a splitting field for a polynomial $f(x)\in k[x]$ and $k\subseteq L$, we have $f(x)\in L[x]$. Thus $K$ is the splitting field of some polynomial over $L$.
In A course in Galois theory, chap. 9, Garling defines $L$ to be a normal extension of $k$ to be an extension which, whenever $f\in k[X]$ is irreducible, either all the roots of $f$ are in $L$, or none.
Then there is a theorem, that states that $L:k$ is normal if and only if $L$ is a splitting field extension for a subset $S\subset k[X]$ (i.e. if every $f\in S$ splits in $L$, and $L$ is the smallest such field). This set is possibly infinite.
Once you have proven this, you can argue as you did even in the infinite extension case (you don't need a unique polynomial to split).