we just learned in class about dense sets in $\Bbb R$ . We learned how to prove that a set is dense in $\Bbb R$ but not how to disprove. I got the following question: Determine whether the set is dense in $\Bbb R$ . $ A =\lbrace a \in \Bbb {R \setminus Q} : a^2 \in \Bbb Q \rbrace$ . I tried proving it but without any success. I also tried disproving it but I think I lack the proper tools. I tried to find two real numbers that don't have any element of A between them.
Any help will be welcomed.
$A$ contains all real numbers of the form $\alpha\sqrt{2}$ where $\alpha\ne 0$ is rational. Take any real number $\beta\ne 0$ and $\gamma=\beta/\sqrt{2}$. The number $\gamma$ can be approximated arbitrarily close by rational numbers $\alpha$. Hence $\beta$ can be approximated arbitrarily close by numbers from $A$. Hence $A$ is dense.