Let's define two recurrence relations with $n \in \mathbb{N}$ and $x \in \left]0 ;1\right[$ as:
$$ \begin{align} a_0 &= 1 \\ a_n &= -x \left( a_{n-1} + n \right) \end{align} $$
and
$$ \begin{align} b_0 &= 1 \\ b_n &= -f(x) \left( b_{n-1} + n \right) \end{align} $$
where $f(x) > x$ if $n$ is odd and $f(x) < x$ if $n$ is even.
Question: How to prove that $a_n > b_n$ if $n$ is odd and $a_n < b_n$ if n is even for all $n \ge 1$?
