Since the Green function $G(x,1)=\sum\limits_{n\in \mathbb{N}_0}P(S_n=x), x\in\mathbb{Z}^d$ gives the expected number of visits to $x$ in a random walk, I'm asked to prove the following:
I have to prove that $G(0,1)=\infty$ if and only if $G(x,1)=\infty$ for any $x\in \mathbb{Z}^d$.
Does this mean that the random walk visits every point in the $d$-dimensional space infinitely many times, and how do I even start to prove this?
You can prove an even more general result: $G(x,1) = \infty$ iff $G(y,1) = \infty$ for any two $x,y \in \mathbb{Z}^d$. For suppose that the random walk visits $x$ infinitely many time. Each time it visits $x$, it has a positive probability $p>0$ to visit $y$ in $d(x,y)$ time steps. Since it visits $x$ infinitely often, this event with probability $p$ will also happen infinitely many times, so it will visit $y$ infinitely often as well.
Of course, this is just the idea of the proof, and for the actual proof you'll have to be a bit more formal.