Given the tower of fields $F_1\subset F_2\subset K$, and $\alpha\in K$ algebraic over $F_1$, can we deduce that (this is not an exercise, and thus more or less a two-fold question: 1. is the statement correct and 2. is the proof correct) $$ [F_2(\alpha):F_2]\leq [F_1(\alpha):F_1] $$?
My attempted proof uses the fact that if $1,\alpha,\alpha^2,\dots,\alpha^k$ are linear dependent in $F_1(\alpha)$ over $F_1$, then they are also linear dependent in $F_2(\alpha)$ over $F_2$.
Note that we assume that all extensions are finite.
The statement is correct, and your idea is fine. Here another proof using that $[F_i(\alpha):F_i]$ is the degree of the minimal polynomial $f_i$ of $\alpha$ over $F_i$.
It is quite clear that $f_2$ divides $f_1$, since $f_1$ is irreducible over $F_1$, but it may split over $F_2$. Thus, $\deg(f_2) \leq \deg(f_1)$.