TITLE: I don't know whether the sequence converges
QUESTION [5 upvotes]: We have $a_0 = x_2\in [0,A]$ and $a_{n+1} = (A−a_n)/2$. Prove that this sequence converges to $A/3$. And then prove the same for the sequences $a_{2k}$ and $a_{2k+1}$.

REPLY [3 votes]: Hint: write $a_n = A/3 + b_n$. Then $$a_{n+1} = (A - a_n) / 2 = A / 3 - b_n / 2 = A/3 + b_{n+1}.$$
Alternative proof: the function $x \mapsto (A - x) / 2$ is a contraction and so by Banach's fixed-point theorem has a unique fixed point $ x = (A - x) / 2$ and so $x = A/3$.