How to do this depends, as Math Major suggests, on exactly how you define the real numbers and what basic properties of the real numbers you can use. I would suggest showing that the set of all real numbers, x, such that

, is non-empty and has, say, 2 as an upper bound. By the "least upper bound" property (which is easily proved by using the Dedekind cut definition of the real numbers), that set must have a least upper bound. Now prove that the least upper bound has the property that its cube is equal to 5.