Martingale not converging a.s.

Question

Let $(X_n)_{n\in \mathbb N}$ be a sequence of independent random variables with $P(X_n = 2^n) = P(X_n = -2^n) = 1/2.$ Let $M_n := \sum_{k=1}^n X_k$. It was easy to show that $M_n$ is a Martingale, but now I want to prove that $M_n$ does not converge a.s. to a real valued random variable $M$. I tried to disprove convergence in probability or that it is unbounded in probability, but feel like I need some help.

Answer 1

$\newcommand{\PM}{\mathbb P}$Hint: \begin{align*} \frac 1 4=\PM(X_n = 2^n , X_{n-1}=2^{n-1}) \leq \PM(M_n \geq 2^n + 2 ) \end{align*} So $M_n$ tends to infinity on a set with strictly positive measure.