Let $f(0) = 0,f(1) = \frac{1}{2}$
and
$$f(n+1) = \frac{f^5(n)}{2} - f(n-1)$$
where $*^5$ is a power.
Then it seems
$$ \sup f(n) = \lim \sup f(n) = \frac{1}{2}$$
and
$$ \inf f(n) = \lim \inf f(n) = \frac{-1}{2}$$
Is this true ?
How to prove it ?
background
Why I wonder about this ?
Well for most values $0<q<1$ I found that
$f(0) = 0,f(1) = q$
and
$$f(n+1) = \frac{f^5(n)}{2} - f(n-1)$$
Then
$$ \sup f(n) = \lim \sup f(n) \neq q$$
and
$$ \inf f(n) = \lim \inf f(n) \neq -q$$
So it seems remarkable that $f(1) = 1/2$ does satisfy it.