
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.

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

No, it's as is written.
Do you think there's a typo?

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

A constant sequence is monotonic. And Converges.

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.

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