The iterates eventually reach the value 1.
The map $U(t)$ takes values in some compact space $G$.
Each factor in (4) has absolute value 1 on $T$.
Let $n_k$ be the first location to the right of the $k$th decimal place of $W$ that has a value less than $b$.
Every element of $A$ has $f$-value 2. [= the value of $f$ at this element is 2]
Then $F$ is less than 1 in absolute value.
The terms of the series (1) decrease in absolute value and their signs alternate.
We claim that, by setting $w$ to zero on this interval, the value of $F(w)$ is reduced.
the largest $k$ value