Let $F$ be a field and let $E$ be the splitting field of a separable polynomial $f \in F[X]$ of degree $n$. Denote by $a_1, \dots, a_n$ the zeros of $f$ in $E$. How do we prove that, for every $i \in \{1,2, \dots, n \},$ $$[F(a_1, \dots, a_i): F] \leq n(n-1)\cdots(n-i+1), $$ and equality happens if $[E:F] = n!$?
I thought about the tower of extensions $$F \subset F(a_1) \subset F(a_1, a_2) \subset \cdots \subset F(a_1, \dots, a_i) \subset \cdots$$ and then at their degrees. As $f$ has degree $n$, it follows that $$[F(a_1):F] \leq n $$ and we can use the line of reasoning to conclude that $$[F(a_1,\dots,a_m):F] \leq (n-m+1).$$ Is this correct? However, for the equality part, I don't know how to proceed.