We will assume and so . Therefore, . We get from here . This was your first half of your problem.
Now, .
(The last inequality follows because )
Suppose that x>0. Define a sequence (s_n) by s_1=k, and s_(n+1) = (((s_n)^2)+x)/(2s_n) for n elements in N. Prove that for any k>0, lim s_n = sqrt(x).
I've gotten that I need to show that (s_(n+1))^2)-x = (((s_n)^2)-x)^2/(4(s_n)^2)=>0, so that s_n=>sqrt(x) for n=>2.
Also I need to prove that (s_n) is decreasing for n=>2, by showing that (s_n)-(s_(n+1))>=0.
Unfortunately I don't know how to exactly show these two steps.
Any help would be great.