By duality and Hahn Banach theorem, we know that for $x\in \ell^1$, its norm can be computed as
$$\|x\|_1=\sup_{\|\beta\|_\infty=1} \left|\sum_k x_k \beta_k\right|.$$
To obtain the norm, in that supremum, is it enough to consider elements $\beta$ such that $\beta_k=\pm 1$?