Write the algebra closure of $ F_p $ as a union of finite fields

In the field theory of Steven Roman Chpater 9 Exercise 20, if we write the algebraic closure of the finite field $ F_q $ as $ Gamma (q) $ Y $ a_n $ any infinite sequence of positive integers that is strictly increased, the exercise wants to prove that $ Gamma (q) = bigcup_ {n = 0} ^ { infty} GF (q ^ {a_n}) $.

However, if $ a_n $ It is an arbitrary sequence, we are even unable to prove $ bigcup_ {n = 0} ^ { infty} GF (q ^ {a_n}) $ it is a field I wonder if the exercise has omitted any condition since the equality is not fulfilled under the current conditions offered.

In fact, I think the condition that $ a_n $ is any sequence of positive integers so that any positive integer $ k $ divide some $ a_n $ It is sufficient and necessary, although I am not sure.

I wait for answers!