Solution
= Solution
<Taylor expansion> of $f'$ about $a_i$ has integral coefficients, so
$$
|f'(a_i+h_i)-f'(a_i)|\le|h_i|=dq_i<d.
$$
In a <Non-Archimedean absolute value>, adding an error of norm strictly smaller than $d$ cannot change a quantity of norm $d$. Therefore
$$
\boxed{|f'(a_i)|=d\quad\text{for every }i.}
$$
This simultaneously verifies the <derivative> assumption used at the next step of part (i).