Exercise 3, § 5

With the notation of Exercise 11 of § 3, let $N$ be the submodule of $\hat{E}$ which is the closure in $\hat{E}$ of the submodule of $\hat{E}$ generated by the vectors $pe_{2n-1} - p^n e_{2n}$ ($n \geq 1$) and let $M = \hat{E}/N$. Show that under the $p$-adic topology the submodule $pM$ of $M$ is not closed in $M$ and deduce that $M$ is a $\mathbf{Z}_p$-module which is ideally Hausdorff but not absolutely Hausdorff with respect to $p$.