a) Let $a>0$ and the sequence $x_n$ fulfills $x_1>0$ and $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ for $n \in \mathbb N$. Show that $x_n \rightarrow \sqrt a$ when $n\rightarrow \infty$.
I have done it in two ways, but I guess I'm not allowed to use the first one and the second one is incomplete. Can someone please help me?
We already know $x_n \rightarrow \sqrt a$, so we do another step of the iteration and see that $x_{n+1} = \sqrt a$.
Using limit, $x_n \rightarrow x, x_{n+1} \rightarrow x$ (this is the part I think it's incomplete, don't I have to show $x_{n+1} \rightarrow x$, how?), we have that
$$x = \frac x 2 (1 + \frac a {x^2}) \Rightarrow 1 = a/x^2 \Rightarrow x = \sqrt a$$
b) Let the sequence $x_n$ be defined as $x_{n+1}= 1 + \frac 1 {x_n} (n \in \mathbb N), x_1=1$. Show that it converges and calculate it's limit.
"Tip: Show that sequences $x_{2n}$ and $x_{2n+1}$ monotone convergent to the limit." I didn't understand the tip, how can this help me? Does it make a difference if the number is odd or even?
Thanks in advance!