In Proposition 1.3.7 of Liu's book, one proves that if a ring $A$ is noetherian then so is $A[[T]]$. We take an ideal $I$ of $A[[T]]$ and prove that there exist $F_1,\ldots,F_m\in I$ such that for all $P\in I$, $n\in \mathbb{N}$ there exist $G_1,\ldots,G_n\in(F_1,\ldots,F_m)$ with $P-\sum_1^n G_i\in\mathfrak{m}^n$ (with $\mathfrak{m}=TA[[T]]$, so $(\mathfrak{m}^n)_n$ is the standard filtration of $A[[T]]$).
For the usual topology on $A[[T]]$ I see that $\sum_1^nG_i$ converges to $P$, but in the proof I need that $\sum_1^\infty G_i$ stays in $(F_1,\ldots,F_m)$ and that I don't see why.
I see that this is a closure property of the ideal $J=(F_1,\ldots,F_m)$ but this don't help me.
I see that we can decompose the sum $\sum_1^nG_i=(A_1^1+\ldots+A_n^1)F_1+\ldots+(A_1^m+\ldots+A_n^m)F_m$ but I don't see why all the sums $A_1^i+\ldots+A_n^i$ should converge.
Liu precises that $G_i$ is in $T^i(F_1,\dots,F_m)$, not merely in $(F_1,\dots,F_m)$ (in fact $T^{i-d}(F_1,\dots,F_m)$ in his notation), hence you can write $$G_1 + G_2 +G_3 + \dots = (A_1^1 T+ A_1^2T^2 + A_1^3T^3 + \dots )F_1 + \dots,$$ which converges in $(F_1,\dots,F_m)$.