I would like to show that an algebraic extension of a perfect field is a perfect field, using the following result:
Given a field $F$ and some family of perfect subfields $\{F_i\}_{i \in I}$ such that $F=\cup _{i\in I} F_i$, we have that $F$ is a perfect field.
EDIT: A perfect field is defined as follows: Any field of characteristic $0$ is perfect, and a field of characteristic $p$ is said to be perfect if any element in $F$ is a $p^{th}$ power of some element in $F$.
I've tried taking some element in the extended field and using the fact that it is algebraic over $F$ in order to construct a perfect subfield that contains the aforementioned element, however couldn't advance much.
Could anyone give me some direction toward the solution?