I am reading https://core.ac.uk/download/pdf/82317236.pdf An introduction to algebraic deformation theory by Thomas F. Fox. On page 23, Theorem 3.1, the author has used some limiting process for sequences of formal power series to conclude the result. In a little more detail,
A non trivial deformation $F=\sum f_it^i$ is given whose infinitesimal $f_n$ is a coboundary. Then there exists a formal isomorphism $F'\cong F$, $F=\sum f'_it^i$ such that $f_m'=0, \forall m < n$. If infinitesimal of $F'$ also turns out to be a coboundary, we can kill it off like above. The paper says this process must stop because F is non-trivial. Why is this so? What if this process continues infinitely?