find the sup of the following sequence

Question

I'm supposed to prove that the following sequence is convergent: $u_0 = 0$, $ u_{n+1} = u_n + \frac{1}{2}(a - u_n^2)$

for $n = 1, 2, ...$ and $a \in [0,1]$

I have already proved that $\forall n$ $ 0\le u_n \le \sqrt a$ and that it is an increasing sequence.

therefore, the sequence is bounded and increasing, it follows that it converges to the sup. I know that $\sqrt a$ is the sup but I am stuck on how to prove it. i'd really appreciate the help

Answer 1

If you already proved the sequence is convergent, then use arithmetic of limits. Suppose

$$\lim_{n\to\infty} u_n =U$$

so that making $\;n\to\infty\;$ in $u_{n+1}=u_n+\frac12(a-u_n^2)$ we get

$$U=U+\frac12(a-U^2)\implies U^2=a\implies U=\pm\sqrt a$$

and as the sequence is positive all the time, we get $\;U=\sqrt a\;$