<s>[INST] Compute limit if it exist of $a_1 = 2$, $a_{n+1} = \frac{1}{2} (a_n + \frac{1}{a_n})$ $$a_1 =  2$$
$$a_{n+1} = \frac{1}{2} (a_n + \frac{1}{a_n})$$
I tried to use induction but it looks as if the sequence isn't monotonic. 
 [/INST] we have $a_1=2$
by induction $2\geq a_n>1$.
$$a_{n+1}=a_n-\frac{a_n^2-1}{2a_n}$$
$$a_{n+1}-1=(a_n-1)(1-\frac{a_n+1}{2a_n})$$
$$=\frac{1}{2}(a_n-1)(1-\frac{1}{a_n})$$
$\implies$
$$|a_{n+1}-1| \leq \frac{1}{2}|a_n-1|$$
$....\leq (\frac{1}{2})^n|a_1-1|$
this proves $(a_n)$ tends to $1$.
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