Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 31D / a / i / Solution

For fixed $n$, define the remaindersThe hypotheses say $r_n=o({\varphi }_n)$ and $r_{n+1}=o({\varphi }_{n+1})$, while subtraction givesBecause $a_{n+1}\ne 0$,For $x$ sufficiently close to $x_0$ this denominator is therefore nonzero, andThus ${\varphi }_{n+1}=o({\varphi }_n)$ for every $n$, so $({\varphi }_n)$ is an asymptotic sequence.