# Monotone Convergence Theorem Question [duplicate]

I'm having trouble with the following problem:

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

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

and

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

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?