If $X_n = 0$ eventually, is $S_n =0$ eventually, where $S_n$ is the sum of $n$ independent rv's with the same distribution as $X_n$?

Question

Let $(X_n)_{n\ge1}$ be an infinite sequence of (not necessarily independent) random variables defined on the same probability space. Suppose that $P(X_n = 0\text{ eventually}) = 1$.

Consider, for every $n\ge1$, $S_n=\sum\limits_{k=1}^nX_k^{(n)}$ the sum of $n$ i.i.d. random variables $X_k^{(n)}$ distributed as $X_n$.

May we conclude that $P(S_n = 0\text{ eventually}) = 1$?

Please note that $S_n$ is not defined as the partial sum $X_1+X_2+\cdots+X_n$.