I am trying to prove that the sequence defined by $x_n = a_n - b_n$ converges to zero where $a_n$ and $b_n$ are sequences such that
$a_n = \sqrt{a_{n-1}b_{n-1}}$ and $ b_n = \frac{a_{n-1} + b_{n-1}}{2}$.
By definition $a_1 = \sqrt{ab}$ and $b_1 = \frac{a + b}{2}$ where $a,b$ are real numbers with $a > b > 0$.
Now I have proven that $x_n$ is strictly increasing, so I just need to show that the supremum of the set of all x_n's is zero. How can I do this?
I tried using contradiction namely to show that $\forall \epsilon > 0$, $\exists n$ such that $-\epsilon < x_n < 0$, but to even produce such an $n$ is tough.
If we complete the square we find that $x_n = -\frac{1}{2}(\sqrt{a_{n-1}} - \sqrt{b_{n-1}})^2 \leq 0.$
That's all I've got at the moment. How can I find the supremum of this?