# Partial limits of recursive sequence

Let $\alpha \in \Re$. Find all the parial limits of the sequence $a_n$ given:

$a_1=\alpha$

$a_{2n}= \frac{a_{2n-1}}{2}$

$a_{2n+1}= \frac12 +a_{2n}$

I've tried to show that $a_n$ has two subsequences $a_{n_{2k}}$ and $a_{n_{2k+1}}$, both monotonic. hence, if I could show monotonicity, there were two possibilities: both limits converge to a finite limit (and then we can find it easily), or both limits tend to infinite.

unfortunately, it seems that that neither the subsequences nor $a_n$ are monotonic. however, by writing down some examples of $a_n$ it seems to converge to $\left (\frac12 \right)$.

Any direction?