Prove using the completeness axiom?

Question

I am trying to figure out how I can prove that there exits $a\in R$ such that $a^3=2$ using the completeness axiom ?

I know that the completeness axiom for the real numbers states that for every non empty set $S$ which is bounded above, then the supremum of $S$ exists.So in the above example, would the supremum be equal to $2$ ?

How do I go about proving the above using the completeness axiom ? Could someone provide a proof for this problem ?

Answer 1

Hint: Consider $S=\{x: x^3<2\}$ it is not empty because it contains $0$, it is bounded above by $3$, show that it sup satisfies $a^3=2$.