Let $k \subseteq K \subseteq E$ fields and if $E/k$ is a finite extension, the $E/K$ and $K/k$ are also finite extensions.
As $E/k$ is finite extension, then $dim_{k}E=n < \infty$. So as, $K \subseteq E$ we have that $K$ is a $k$-vectorial subspace of $E$. And by a well known result in linear algebra we have that $dim_{k}K \leq dim_{k}E=n=\infty$. Proving that $K/k$ is also a finite field extension.
Now I want to prove that; $E/K$ is a finite field extension. By some field theory result it should happen that if $E/K$ is a finite field extension
$$(dim_{k}K) (dim_{K}E)=dim_{k}E.$$
So if I suppose $E/K$ is an infinite field extension, then
$$(dim_{k}K) (dim_{K}E)=(dim_{k}K)(\infty)=dim_{k}E=\infty.$$
Then, $dim_{k}E=\infty=n < \infty$. Proving that $E/K$ is a finite field extension.
Is my proof right? If not what Im suppose to correct? Also If this proof right would appreciate to see another solutions for this problem. Thanks!
There's some awkwardness, where you say things like $n=\infty$ and $\infty=n<\infty.$
The core of the approach is simply to note that $$\left(\operatorname{dim}_kK\right)\left(\operatorname{dim}_KE\right)=\operatorname{dim}_kE.$$ Since $\operatorname{dim}_kE$ is finite, and since $\operatorname{dim}_kK$ and $\operatorname{dim}_KE$ are either positive integers or infinite, then they must be positive integers, so both $E/K$ and $K/k$ are finite extensions.