How do you layout the proof for $\forall \in \Bbb R, \exists \in \Bbb R ∶ − ^3 = 0$ being true?

Question

So I know it's true and I understand why it's true but I don't understand how I'm supposed to give the answer, as the answer to this practice question is the following:

How exactly am I supposed to layout the answer for this?

Answer 1

How exactly am I supposed to lay out the answer for this?

Let $a$ be an arbitrary real number.

Then put $y=\sqrt[3]a$ so that $y$ is real and $$a-y^3=0.$$ That is, there exists a real $y$ for which $a-y^3=0.$

As $a$ is arbitrary, hence

• for each real $x,$ there exists a real $y$ for which $x-y^3=0.$

as required.

This sample solutution can be condensed, or even expanded, as preferred.