Proposition 3

Let A be a filtered ring, (An) its filtration, B the subring $\bigcup_{n \in \mathbf{Z}} A_n$ of A, E a filtered B-module, (E_n) its filtration and F the sub-B-module $\bigcup_{n \in \mathbf{Z}} E_n$ of E. Then the mapping $(a, x) \mapsto ax$ from $B \times F$ to F is continuous.*

Let $a, \in B, x_0 \in F$; there exists by hypothesis integers $r, s$ such that $a_0 \in A_r$ and $x_0 \in E_s$. The relation

shows that if $a - a_0 \in A_i$ and $x - x_0 \in E_j$, $ax - a_0x_0$ belongs to

Then, given an integer $n$, $ax - a_0x_0 \in E_n$ provided $i \geq n - s, j \geq n - r$ and $i + j \geq n$, that is so long as $i$ and $j$ are sufficiently large.