What is the limit of this sequence? $x_n=\frac 13(x_{n-1} + \frac{6}{x_{n-1}})$, $x_0=2$

Question

$x_n=\frac 13(x_{n-1} + \frac{6}{x_{n-1}})$

Where $x_0 = 2$

I've calculated enough values to see that the limit of $x_n$ for $n$ to infinity is $\sqrt 3.$

But how do I prove this beside observation?

Answer 1

First prove that it is bounded below by 1 and bounded above by 3 using induction and again by induction show that it is monotonic decreasing. It confirms the existence of limit and then it follows limit is $\sqrt3$.