Exercise 3, § 1

Let E be an A-module.

(a) Let $(L_\mu)_{\mu \in M}$ be a family of right directed ordered sets and, for all $\mu \in M$, let $(F_{\lambda \mu})_{\lambda \in L_\mu}$ be an increasing family of submodules of E. Show that

(b) Take A to be a field and E the A-module $A^N$; let G be the submodule $A^{(N)}$ of E; on the other hand, for every finite subset H of $\mathbf{N}$, let $F_H$ be the submodule $\prod_{n \in \mathbf{N}} M_n$ of E, where $M_n = A$ for $n \notin H$ and $M_n = 0$ for $n \in H$. Show that the family $(F_H)$, where H runs throughout the set $\Phi$ of finite subsets of $\mathbf{N}$ is right directed and that