ELMO SL 2017 A1

Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by

$$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$

for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$.

Proposed by Michael Ma