1. ## Monotone sequence

Given $\displaystyle y_n=\dfrac12(b+y_n^2)$ with $\displaystyle y_0=0.$

Show that $\displaystyle y_n$ is monotone.

I'm getting trouble to do that. It's a part of another problem which says to prove that $\displaystyle y_n$ converges, so in order to do that, I already proved that $\displaystyle y_n$ is bounded below.

2. Do you mean $\displaystyle y_{n+1}=\dfrac12(b+y_n^2)$?

3. No, it's as is written.

Do you think there's a typo?

4. There must be. Otherwise the sequence is constant for n>0.

5. A constant sequence is monotonic. And Converges.

6. This particular sequence may or may not be monotonic. The given equation may have 2 solutions (depending on b), and therefore the given sequence isn't well defined - it could take on one of these values, or bounce between the 2. In any case, it is more likely that the problem has a typo.

7. yn is either a real or complex constant depending on b. in either case its constant, monotonic (absoluteley if b complex), and convergent.