Exercise 3, § 6

Let $(F_n, f_{mn})$ be the direct system of $\mathbf{Z}$-modules such that $F_n$ is equal to $U_p$ (§ 1, Exercise 12 (d)) for all $n \geq 0$ and $f_{mn}$ is the endomorphism $x \mapsto p^{m-n}x$ of $U_p$ for $n \leq m$. Show that $\lim \rightarrow F_n = \{0\}$ although the $f_{mn}$ are surjective.