I should think so. Given a real sequence $\{a_{n}\} \to 0$ and an arbitrary $\epsilon > 0$, we have that there exists some $N$ such that for all $n > N$, $|a_{n}| < \epsilon$. That being said, let $\sigma: \mathbb{N} \to \mathbb{N}$ be a permutation, and consider the rearranged sequence $\{a_{\sigma(n)}\}$. Since the convergence of the initial sequence implies that all but finitely many elements in the sequence can be more then $\epsilon$ away from $0$, after a finite number $N'$ of terms in this new, rearranged sequence, we will have exhausted all those cases, and for $n > N'$ we will have $|a_{\sigma(n)}| < \epsilon$. Note that this works with $0$ replaced by an arbitrary finite limit $L$.
Note that this fails for series, as a result of the Riemann rearrangement theorem.