$F(a_1,\ldots,a_n) = F(a_1+t_2 a_n+\cdots+t_n a_n)$ if Extension is Simple?

Question

Suppose that F is an infinite field and that $L = F(a_1,\ldots,a_n)$ is a simple extension of $F$.

Can you show, without the assumption that $[L:F] < \infty$ that $\exists t_2,\ldots,t_n \in F$ such that $L = F(a_1+t_2 a_n+\cdots+t_n a_n)$?

With that assumption I would use the fact that, under these assumptions, there are a finite number of subfields, and find $\beta_1,\beta_2\in F$ s.t $F(a_1+\beta_1 a_2) = F(a_1+\beta_2 a_2)$ (since $F$ infinite), then conclude that $a_1,a_2 \in F(a_1 + \beta_1 a_2)$. From here proceed by induction on $n$.

Thanks!