Let $F\subset L\subset K$ be fields.
Seen $K$ as a vector space over $F$, if we set $[K:F]$= dimension of $K$ over $F$, Is it true that if $[K:F]< \infty$, then $[L:F]< \infty$?
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Let's write $n = [K:F]$ (which is finite here).
Then $K$ is a $F$- vector space of dimension $n$. Now, you can easily check that $L$ is a $F$-subspace of $K$. Now, just use the common result that a subspace of a finite-dimensional vector space is itself a finite-dimensional subspace to conclude.