I have the following exercise:
Let $a_0 > b_0 > 0,$, with $ a_0, b_0 \in \mathbb{R}$. We define $a_n$ and $b_n$:
$ a_{n+1} = \frac{a_n+b_n}{2} $ $ b_{n+1} = \sqrt{a_n b_n} $
Prove the following:
- $a_n \ge b_n \forall n$.
- $a_n$ is decreasing and $b_n$ is increasing.
Question 2 is simple enough, it's question 1 I'm having trouble with. I tried to use induction: I first show that for $n= 0$ the inequality is true, and then, assuming $a_n \ge b_n$, I want to show that $a_{n+1} \ge b_{n+1}$, i.e., $\frac{a_n+b_n}{2} \ge \sqrt{a_n b_n}$
I have tried for a while but haven't managed to get anything. Does anyone have any suggestions?