Prove the following sequence is convergent: $a(1) = 1$ and $a(n+1) = 1/2 \cdot (a(n) + 3/a(n))$ I know the limit is $\sqrt 3$. And I tried proving that the sequence is decreasing from $a(2)$, yet when I got to this point, I have been stuck:
$a(n+1) - a(n) = (3 - a(n)^2) / 2a(n)$ ?
Thank you very much!

Hint:
On the interval $(\sqrt 3,+\infty)$ the function $\:f(x)=\dfrac12\Bigl(x+\dfrac 3x\Bigr)$ is increasing and $\:f(x)<x$.
Note that $a_2=2\in(\sqrt 3,+\infty)$.
