Suppose we have a sequence of iid random variables: $(X_n,n\geq 1)$ such that $P(X_n=1)=(X_n=-1)=1/2$.
I managed to prove that the process defined as: $$ M_0= x\quad 0<x<1 ,\quad M_{n+1} = M_n + \frac{X_{n+1}}{2^{n+1}}$$ is a martingale. I should check now if it converges to a limit almost surely and if there exists a $M_{\infty}: M_n = E(M_{\infty}|\mathcal{F}_n)$. In order to do so I am trying to show that $\sup_n E(M_{n+1}^2) < \infty$. It should hold that $E(M_{n+1}^2) = E(M_n^2) + \frac{1}{2^{2(n+1)}} $. Following this I would say that for every $n\in\mathbb{N}, E(M_n^2) = \sum_{i=1}^n \frac{1}{2^{(2n)}} + x^2 = <\infty$.
Thus, the martingale should be such that $\sup_nE(M_n^2)<\infty$ and $M_{\infty}$ exists. Is it right? Thanks!
Actually it holds $$E(M_n^2) = E(M_n)^2 + \frac{1}{4^{n+1}}$$ and you can directly note what $$\sup_{n\in\Bbb N} E(M_n^2)$$ is to prove it's not infinity. Just because $$E(M_n^2) < \infty$$ holds it does not necessarily follow that $$\sup_{n\in\Bbb N} E(M_n^2) < \infty$$ as well. OFC it does here but you should mention this by either give the value of it or give an estimation. The rest seems to be fine here.