1. Convergence Proof

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

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

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

No clue where to start...

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

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