About particular extension field

Question

Let $E/K$ be an extension field. If $A\subseteq E$ and $u\in K(A)$, prove that there are $a_1,\dots,a_n\in A$ such that $u\in K(a_1,\dots,a_n)$.

I thought to use the fact that $K(A)$ is a $K(u)$-vector space, but I didn't manage to progress much. Please let me know if this is a duplicate question.

Answer 1

Consider the set $L=$ {$x$ | $x$ can be described with rational expression with $k_1,..,k_n∈K$ and $a_1,..,a_m∈A$}

You can verify that:
・$L$ forms a field.
・Field containing $K(A)$ must contain $L$

So $L = K(A)$ and its every element can be described with rational expression with $k_1,..,k_n∈K$ and $a_1,..,a_m∈A$.