I'm trying to understand the this proof of the generalization of Hilbert's basis theorem to formal power series. Please, don't mark this as a duplicate since I don't ask for a general proof of this fact, but rather want to explain a step in a specific proof to me. For convenience, I also add a screenshot to this post.
I don't understand the last part of the proof after the "so on". Suppose that we know that, for each $n \geq 0$, there are $c_{1,n},...,c_{n_{d_0},n} \in R$ such that $$f - \sum c_{d,i}f_{d,i} - \sum c_{i,1}xf_{d_0,i} - ... - \sum c_{i,n}x^nf_{d_0,i} \in (x^{d_0 + n + 1})\cap I.$$
How this implies that
$f = \sum c_{d,i}f_{d,i} + \sum_i(\sum_e c_{i,e}x^e) f_{d_0,i}$? We can't sum infinitely many factors in a ring, even if it is a ring of power series.
You can define convergence in the ring of formal power series $R[[x]]$. The sequence of ideals $J_n:=(x^n)$, $n\geq 0$, defines a neighborhood of $0$. The function $v(f) = \max\{n : f\in J_n\}$ (and $v(0)=\infty$) defines a metric $d(f,g)=2^{-v(f-g)}$. Hence the infinite sums are well-defined (they converge in the above sense).
In the sum $$ f = \sum c_{d,i}f_{d,i} + \sum_i(\sum_e c_{i,e}x^e) f_{d_0,i} $$ the only "infinite" parts are the power series $\sum_ec_{i,e}x^e$; the sums over $i,d$ are finite, and $f\in I'$.