Exercise 1, § 6

Let $(E_n, f_{nm})$ be the inverse system of $\mathbf{Z}$-modules whose indexing set is $\mathbf{N}$, such that $E_n = \mathbf{Z}$ for all $n$ and, for $n \leq m$, $f_{nm}$ is the mapping $x \mapsto 3^{m-n}x$. For all $n$, let $u_n$ be the canonical mapping $E_n \to \mathbf{Z}/2\mathbf{Z}$. Show that $(u_n)$ is an inverse system of surjective linear mappings but that $\lim \leftarrow u_n$ is not surjective.