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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?

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    Why don't you let $=$ and $=$2012-12-02
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    @Amr is equivalent to but may be not equal.2012-12-02
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    oh i see. I thought you were using = as equivalent2012-12-02
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    Does this not follow immediately from the transitivity property for equivalence relations?2012-12-02

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