Infinite algebraic field extension of a finite field is normal and separable

Question

I am trying to prove that every infinite algebraic field extension $K$ of a finite field $\mathbb{F}$ is separable and normal. I know how to do it for finite $K$, but I am struggling to see why the infinite case is also true. I have seen the answer here, but it does not help. I think I should prove that $K$ is an infinite union of finite fields $\mathbb{F}_{p}$, where $p$ is prime, but I do not know how to do it.