Show that the simple symmetric random walk on $\mathbb{Z}$ is recurrent.

Three particles perform independent simple symmetric random walks on $\mathbb{Z}$. What is the probability that they are all simultaneously at 0 infinitely often? Justify your answer.

[You may assume without proof that there exist constants $A, B>0$ such that $A \sqrt{n}(n / e)^{n} \leqslant n ! \leqslant B \sqrt{n}(n / e)^{n}$ for all positive integers $\left.n .\right]$