Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $ a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .
Nice and easy.
Note that if $\forall n, b_n <0$ then we can replace it with $-b_n$ so assume WLOG, $b_n>0 \forall n>M$
moreover note that if $b_n<b_{n+1}$ for some $n$ then we are done. Else, assume it to be strictly increasing and note that the given condition means that the sequence $x_n =b_{n+1}-b_n$ is decreasing and bounded above.
This implies that $b_n < c(2-\frac{1}{a_{n-1}}) \forall n$ and so is bounded. ($a_n<1$ otherwise we are done. thought i mention it)
This is wrong as pointed by PRO2000.