Proving convergence of a sequence given by $a_{n+1}=A+Ba_n^3$

Question

Given $A,B>0, A+B=1, 0<a_{1}\le A$ and a sequence $(a_n)$ with $a_{n+1}=A+Ba_n^3$. Determine with proof whether the sequence $(a_n)$ is convergent. I have tried a few values of $A,B$ and $a_1$, and found that the sequence is convergent. However, I could not prove it rigorously. My approach to this question is using Monotone Convergence Theorem. I have managed to prove that the sequence is bounded above, but failed to prove that it is increasing. Am I on the right track? Or is there any other method to solve this question? Thanks for the help!

Answer 1

You can show that the sequence is increasing via induction:

We have $a_2 > A \geq a_1$, so the start works. Now let $n>1$. We assume that we already showed that $a_n > a_{n-1}$. We have

$$ a_{n+1} = A+B a_n^3 > A+B a_{n-1}^3 = a_n $$ and we are done.