Let $\langle a_n\rangle$ , $\langle b_n\rangle$ , $\langle c_n\rangle$ be Cauchy sequences of rational numbers, and $\langle c_n\rangle$ is equivalent to $\langle a_nb_n\rangle$. Prove or disprove that there are two Cauchy sequences $\langle a_n'\rangle$ , $\langle b_n'\rangle$ of rational numbers such that
(1) $\langle a_n\rangle$ is equivalent to $\langle a_n'\rangle$ ;
(2) $\langle b_n\rangle$ is equivalent to $\langle b_n'\rangle$ ;
(3) $\langle c_n\rangle=\langle a_n'b_n'\rangle$ .
If it is true, can we prove it intuitionistically?