Existence of Intermediate field $F$ of $E$ and $G$ such that the $[G: F] < \infty$ if $G$ is an infinite extension of $E$
$$E \subset G $$
Then does there exists an extension $F$ of $E$ such that $G$ is finite extension of $F$
$$E \subset F \subset G$$
I think we need to remove some kind of elements from $G$ to get $F$, but obviously it looks silly as it has not been defined. Hence I need help here.

If we require $G\ne F$ then there does not necessarily exist such a field $F$. Take for example $G=\Bbb R$ and $E=\Bbb Q$. If $F$ is a subfield of $\Bbb R$ with $[\Bbb R:F]<\infty$, then also $[\Bbb C:F]<\infty$. However, as $\Bbb C$ is algebraically closed, the Artin-Schreier theorem implies that $F=\Bbb R$. Hence there does not exist such a field with $F\ne G$.
Edit: For completeness, here is the statement of the theorem:

Let $L$ be an algebraically closed field and $K$ a subfield with $[L:K]<\infty$. Then $L$ is obtained from $K$ by adjoining a squareroot of $-1$, in particular $[L:K]=1$ or $2$.

For a reference, see e.g. here.
