Proposition 8

Suppose that A and E satisfy the hypotheses of Proposition 7. Let $(F_p)$ be a decreasing sequence of closed submodules of E such that $\bigcap F_p = 0$. Then, for every neighbourhood V of 0 in E, there exists $p$ such that $F_p \subset V$ (in other words, the base of the filter $(F_p)$ converges to 0).

We may assume that V is one of the $E_n$, in which case $E/V$ is Artinian. Let us write $F'_p = (F_p + V)/V$; as the $F'_p$ form a decreasing sequence of submodules of $E/V$, there exists an integer $j$ such that $F'_p = F'_j$ for all $p > j$. We shall see that $F'_j = \{0\}$, which will complete the proof. Let $x \in F'_j$ and let W, be the set of elements of F, whose image in $E/V$ is $x$ ($p \geq j$); by definition of j, the W, are non-empty closed affine subsets of E and obviously $W_{p+1} \subset W$;

then it follows from Proposition 7 that there exists an element y belonging to all the W,. As W, $\subset F,$ and $\bigcap_{p \in \mathbf{N}} F_p = \{0\}, y = 0$; since x is the canonical image of y in E/V, $x = 0$ (cf. Exercises 15 to 21).