I am showing that $\left(\forall x\in\mathbb R\right)\left(\exists y\in\mathbb R\right)$ s. t. $y^3 = x$.
I begin with the case where $x > 0$ and consider the set $E := \{z | z \in \mathbb R, z^3 < x\}$ To be rigorous, is it sufficient to say that $\sup E =\sqrt[3]{x}$ based on the inequality defining the set $E$?
I know that if this is the case, then we can simply invoke the completeness axiom to show that $\sup E\in\mathbb R$ and define $y = \sup E$.