Prove that the sequence $a_{n+1} =frac{1}{2}left(a_{n}+frac{c}{a_{n}}right)$ is convergent and find its limit

Let $c>0$, $a_{1} = 1$, and
$$a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$$

I need to:

1. Show that $a_{n}$ is defined for every $n\geq 1$


2. Show that this sequence is convergent.


3. Find its limit.

I proved the first part by showing by induction that this sequence is positive for every $n$. To show that this sequence is convergent I'm thinking of showing that this sequence is a Cauchy series, yet can't figure out how.

For the third part I'm clueless at the moment.

Answer

Hints:

1. One can prove that if $a_1>0$ and $n \geq 2$ then $a_n \geq \sqrt{c}$.


2. One can prove that if $a_1>0$ and $n \geq 2$ then $a_{n+1} \leq a_n$. Knowing that and the bound in 1, convergence follows.


3. Once you know that a recursive sequence is convergent, its limit can only be a fixed point of the recursion mapping, i.e. in your case a solution to $\frac{x}{2}+\frac{c}{2x}=x$.