I've been having trouble with this homework problem for some time. I believe that I need to prove the identity statements by induction, but I'm not sure what that would look like. Similarly, I have little idea how to tackle the limit proof as I don't know how to prove convergence without an explicit formula for the sequence. In general, any hints or suggestions on how to approach this problem would be greatly appreciated.
Fix a > 0 and iteratively define a sequence $(x_{n})_{n\in N}$ as
$$x_1 = a, \;\;\;\; x_{n+1} = \frac{1}{2}(x_{n} + \frac{a}{x_n}) \;\;\;\;\;\text{ if } n \geq 1.$$
- Show that the sequence $(x_n)_{n \in N}$ satisfies for any $n \in \mathbb{N}$ the identities
$$ \frac{x_{n+1} - \sqrt{a}}{x_{n+1} + \sqrt{a}} = \Big( \frac{x_n - \sqrt{a}}{x_n + \sqrt{a}}\Big)^2, \qquad x_{n} = \sqrt{a} \Bigg( \frac{1 + \frac{x_n - \sqrt{a}}{x_n + \sqrt{a}}}{1 - \frac{x_n - \sqrt{a}}{x_n + \sqrt{a}}}\Bigg). $$
- Prove that the sequence $(x_{n})_{n\in N}$ converges. What is the limit?
Any input on how to handle to either portion of the problem would be tremendously helpful.
