Given $X_1,X_2,\dots$ i.i.d. $\mathbb{R}^d$-valued random variables, define $S_0 := 0$, $S_n := X_1 + \dots + X_n$ to be a random walk starting at $0$. Suppose that for some $x \in \mathbb{R}^d$ there exists $n$ such that $\mathbb{P}(S_n = x) > 0$.
Suppose that $S_n$ is recurrent (resp. transient) at $0$ (so with probability one, $S_n$ returns infinitely (resp. finitely) many times to $0$). Show that with probability one, $S_n$ visits $x$ infinitely (resp. finitely) many times.
It seems like this result should follow easily from the translation/time invariance of a random walk, but I'm having trouble writing a concrete proof. Any tips on how to tackle it?