 Solution

Since $K\subseteq L$ and $L/F$ is finite, an $F$-linearly independent set in $K$ remains independent in $L$, so $K/F$ is finite. A finite $F$-spanning set of $L$ also spans it over $K$, so $L/K$ is finite.

Take an $F$-basis $a_1,\dots ,a_m$ of $K$ and a $K$-basis $b_1,\dots ,b_n$ of $L$. Expanding coefficients twice shows that all products $a_i{b}_j$ span $L$ over $F$. If ${\sum }_{i,j}{c}_{ij}{a}_i{b}_j=0$ with $c_{ij}\in F$, independence of the $b_j$ forces ${\sum }_i{c}_{ij}{a}_i=0$ for each $j$, and then independence of the $a_i$ forces every coefficient zero. Thus these products form an $F$-basis, proving