Does a symmetric random walk converge almost surely?

Question

Does a symmetric random walk in- one dimension- converge almost surely? Can we prove or disprove it by martingales?

Answer 1

No, it diverges almost surely, in the sense that for all $n\in\mathbb N$ and all $M\in\mathbb N$ there is almost surely an $n_+\gt n$ such that $X_{n_+}\gt M$ and an $n_-\gt n$ such that $x_{n_-}\lt-M$.