Prove that $a_n$ converges: $a_1 = 0$ , $a_2 = 1/2$, $a_{{n+1}} = (1/3)(1+a_n+(a_{n-1})^2)$
My game plan is to prove that this sequence is monotonic rising and then bounded to prove convergence, but I can't even prove that its monotonic rising. (My question is relating to first proving that the sequence is monotonic so please don't give me the answer for the rest of the proof I want to still try it myself)
I'm trying to prove this by induction. I've found the first few $n$'s : $(n_1 = 0,n_2 = 1/2,n_3= 1/2,n_4= 7/12)$ so it looks like this is indeed rising. I've decided to assume that for $n=k$, $a_{{k+1}}>a_{{k}}$ and to show that it's rising for all $k$'s and therefore all $n$'s. Not sure what to do now. I've searched online and on youtube for examples on proving convergence for recursive sequences but only found very trivial examples of induction (a method I admittedly don't completely understand)
again, don't post how to prove the convergence I want to try to figure this out still.
Thanks, nofe