Topology on two dimensional local fields

I am reading the book of Schneider about Galois representation and $(\varphi, \Gamma)$-module, Section 1.7, but I don't understand his proof on Lemma 1.7.6. In this section, he introduced the weak topology on the two dimensional local field $\mathscr{A}_L$, where $L$ is a local field, with $\pi$ is a uniformizer of L. $\mathscr{A}_L$ can be identified with the ring of infinite Laurent series, with coefficients go to $0$ when the indexes go to $-\infty$. Let $k_L$ be the residue field of $L$, then the field $k_L((X))$ is the residue field of $\mathscr{A}_L$.

As I understand, we want to equip the topology on $\mathscr{A}_L$ such that the projection map from $\mathscr{A}_L$ to $k_L((X))$ is continuous, so we can choose a system of neighborhoods near $0$ (on $\mathscr{A}_L$) defined by

$$U_m := X^m O_L[[X]] + \pi^m \mathscr{A}_L$$

Next, in Lemma 1.7.6, he proved that with this topology, $\mathscr{A}_L$ is Hausdorff and complete, but I don't understand the proof. After checking the inverse limit identity, he says "it is sufficient to prove the multiplication map is continuous". Why does it imply the statements?

There is a paper of Madunts and Zhukov "Multidimensional complete fields: Topology and other basic constructions", which probably contains the answer but I cannot access. Could anyone help?

• Crossposted at MO. – Dietrich Burde Dec 28 '17 at 20:13