= Solution
The <cobweb plot> uses the <curve> $v=\sqrt{2+u}$ and the diagonal $v=u$. From a point on the diagonal, a vertical step to the <curve> computes the next iterate; a horizontal step returns its value to the diagonal.
For $0\leq u<2$, $(2+u)-u^2=(2-u)(1+u)>0$, so $u<\sqrt{2+u}<2$. Thus a start below $2$ gives a strictly increasing <sequence> bounded above by $2$. For $u>2$ the <inequalities> reverse to $2<\sqrt{2+u}<u$, giving a strictly decreasing <sequence> bounded below by $2$. Both types of staircase approach the same intersection.
The <bounded monotone sequence theorem> proves existence of a limit, and <continuity> then makes it a nonnegative <fixed point>. Part (i) gives
$$
\boxed{\lim_{n\to\infty}y_n=2\quad\text{for every }y_0\geq0.}
$$
This proves the <global convergence of a square-root iteration>; the graph below illustrates both cases without relying on a finite numerical sample as proof.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1-square-root-cobweb.png]
{title=Iteration staircases approaching the positive fixed point from below and from above}
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