I was reading Georgi Shilov's Elementary Real and Complex Analysis. In section 6.48 (p.209), he proves that on $\mathbb R^n$, if a two-sided series $\sum_{k=-\infty}^\infty a_k$ converges, i.e. if there is a vector $s\in\mathbb R^n$ such that for any $\epsilon>0$, there exists an integer $N>0$ that makes $\left\|s-\sum_{k=-p}^qa_k\right\|<\epsilon$ for any $p,q\ge N$, then the one-sided series $\sum_{k=1}^\infty a_k$ and $\sum_{k=1}^\infty a_{-k}$ also converge.
His proof makes use of Cauchy sequence. Is it possible to infer the convergences of the one-sided series without using the completeness of $\mathbb R$?