Exercise 5, § 6

Let $(C^{(m)})_{m \geq 0}$ be a sequence of sieves $C^{(m)} = (C_n^{(m)}, p_n^{(m)})_{n \geq 0}$. For every $n \geq 0$, put

where $a_m = 0$ for all $m \geq 0$; the mapping $p_n : D_{n+1} \to D_n$ is taken to be equal to where $q^{(n+1)}$ is the unique mapping from $C_0^{(n+1)}$ onto $\{0\}$, and $\mathrm{id}_m$ is the identity map of $\{a_m\}$. Then $(D_n, p_n)$ is a sieve.

Let $(E^{(m)})_{m \geq 0}$ be a sequence of Hausdorff locally convex spaces; we assume that for each $m$ there exists an exhaustion $(C_n^{(m)}, p_n^{(m)}, \phi_n^{(m)})_{n \geq 0}$ of $E^{(m)}$. Consider the Hausdorff locally convex space $E = \prod_m E^{(m)}$, and for every $n$, put

where $\psi_m$ is the mapping from $\{a_m\}$ into the set of convex and balanced subsets of $E^{(m)}$ such that $\psi_n(a_m) = E^{(m)}$. Show that $(D_n, p_n, \phi_n)$ is an exhaustion on the product space E.