The Stacks project

Exercise 111.2.5. Let $(I, \geq )$ be a directed set and let $(A_ i, \varphi _{ij})$ be a system of rings over $I$ with colimit $A$. Suppose that $\mathop{\mathrm{Spec}}(A_ j) \to \mathop{\mathrm{Spec}}(A_ i)$ is surjective for all $i \leq j$. Show that $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A_ i)$ is surjective for all $i$. (Hint: You can try to use Tychonoff, but there is also a basically trivial direct algebraic proof based on Algebra, Lemma 10.18.6.)