Exercise 29, § 6

a) Now prove that there exists a double sequence $(E_{nm})$ of finite symmetric subsets of $\mathbf{Z}$ defined for $0 \leq n \leq m$, containing 0, not reduced to 0 for $m \geq 1$, and satisfying the following conditions: 1° for every $n \geq 0$, the sets $E_{n+1,m} + (\bigcup_{r=n+1}^m E_{n+1,r})$, $E_{n+1,m} \cdot (\bigcup_{r=n+1}^m E_{n+1,r})$ and $k \cdot E_{n+1,m}$ for $k \leq n+1$ are all contained in the union $\bigcup_{s=n}^m E_{ns}$; 2° if $y_m$ denotes the smallest element $> 0$ of $E_{0,m} - \{0\}$, and $z_m$ the greatest element of $E_{0,m}$, one has $y_{m+1} \geq z_m + m + 1$; 3° $E_{n+1,m} \subset E_{n,m}$. (Supposing the $E_{pq}$ determined for $p \leq q \leq m$, determine successively $E_{m+1,m+1}, E_{m,m+1}, \ldots, E_{0,m+1}$ by descending recurrence).

b) Let $V_n$ denote the union of the $E_{nm}$ for all $m \geq n$. Now prove that $(V_n)$ is a fundamental system of neighbourhoods of 0 for a topology on $\mathbf{Z}$ compatible with the ring structure of $\mathbf{Z}$, and for which no subgroup of $\mathbf{Z}$ other than $\mathbf{Z}$ is open.