1. ## Convergence Proof

Suppose that A > 0 is a given real number. Define the sequence {$\displaystyle a_n$}$\displaystyle n \in N$ recursively by:

$\displaystyle a_1$ is a positive real number; $\displaystyle a_{n+1} = \frac{1}{2} (a_n + \frac{A}{a_n})$.

Prove that {$\displaystyle a_n$}$\displaystyle n \in N$ converges to $\displaystyle \sqrt{A}$

No clue where to start...

2. Originally Posted by thaopanda
Suppose that A > 0 is a given real number. Define the sequence {$\displaystyle a_n$}$\displaystyle n \in N$ recursively by:
$\displaystyle a_1$ is a positive real number; $\displaystyle a_{n+1} = \frac{1}{2} (a_n + \frac{A}{a_n})$.
Prove that {$\displaystyle a_n$}$\displaystyle n \in N$ converges to $\displaystyle \sqrt{A}$
Use induction to prove the sequence becomes decreasing bounded below.
From that you know that the sequence converges to say $\displaystyle L$.
Then solve $\displaystyle L=\frac{1}{2}\left(L+\frac{A}{L}\right)$ for $\displaystyle L$

3. thanks! that was a lot easier than I thought