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I'm having trouble with the following problem:

Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences satisfying

$$a_{n + 1} = \frac{a_{n} + b_{n}}{2}$$


$$b_{n + 1} = \sqrt{a_{n}b_{n}}$$

Show that the sequences are both convergent, and they converge to the same limit.

So, someone else's solution proves that $a_{n} \geq a_{n + 1} \geq b_{n + 1} \geq b_{n}$, and I understand everything up until there. But then, they say that $a_{n} \geq a_{n + 1}$ implies $\{a_{n}\}$ is monotonically decreasing, and $b$ is a bound for $\{a_{n}\}$, so by the Monotone Convergence Theorem, the sequence converges.

But, I thought that we cannot say $b$ is a bound for $\{a_{n}\}$ since that would be assuming that $\{b_{n}\}$ converges, which is part of what we want to show? After stating $\{a_{n}\}$ converges, they claim $b_{n + 1} \geq b_{n}$ implies $\{b_{n}\}$ is monotonically increasing, and $a$ is a bound for the sequence, so it converges.

Is this proof fine?


marked as duplicate by Mike Earnest, Lord Shark the Unknown, Jack D'Aurizio real-analysis Sep 24 '18 at 19:19

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