Surjectivity of projective maps from an inverse limit to an element of the direct product

Question

I'm doing some self-study and I'm stuck on a problem involving inverse/projective limits. Although this is NOT a homework problem, I'd really appreciate some hints rather than a completely worked out solution.

The problem is from Dummit & Foote:

I'm having a rather difficult time figuring out how to prove part (b).

I think I need to establish that for every $a_{i}\in A_{i}$, there exists some

$$\alpha=(\,\alpha_{1},\alpha_{2},\ldots,\alpha_{j},\ldots)\in\prod_{i\in I}A_{i}$$

such that $\alpha\in P$ and $\mu_{i}(\alpha)=a_{i}$. I'm just not really sure how to do that.

Answer 1

Let $a_m \in A_m$. To get what you want, you need to extend this to a sequence $(a_1, a_2, a_3, \ldots)$ such that $a_i = \mu_{j,i} (a_j)$ for all $i \le j$. Given the hypotheses, there's really only one way to proceed: by induction. If you're worried about set-theoretic details, then I will also point out that you will need the axiom of dependent choice.