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Thread: Cauchy equivalence relation question

  1. #1
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    Cauchy equivalence relation question

    Show that if $\displaystyle \{a_n\}$ and $\displaystyle \{b_n\}$ are Cauchy in $\displaystyle \mathbb{Q}$ contained in equivalence classes $\displaystyle a$ and $\displaystyle b$ (real numbers) respectively, and if $\displaystyle a_n < b_n$ for all $\displaystyle n$, then there exists a real number $\displaystyle \delta$ which contains a rational Cauchy sequence $\displaystyle \{\delta_n\}$ each of whose $\displaystyle \delta_n$ is positive and $\displaystyle b=a+\delta$.

    Here's what I have, for some reason it doesn't seem correct.

    since $\displaystyle a_n < b_n$ then $\displaystyle 0 < b_n-a_n$ implying that there exist $\displaystyle \delta_n$ such that $\displaystyle \delta_n = b_n-a_n$ for all $\displaystyle n$ but since $\displaystyle {\delta_n}$ is an equivalence class of a Cauchy sequence then $\displaystyle [\{\delta_n\}] = \delta \in \mathbb{R}$ and the same for $\displaystyle [\{a_n}]=a \in \mathbb{R}$ and $\displaystyle [\{b_n}] = b \in \mathbb{R}$ so then $\displaystyle [\{\delta_n\}] = [\{b_n\}]-[\{a_n\}]\Rightarrow \delta = b-a \Rightarrow a+\delta = b$
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  2. #2
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    Quote Originally Posted by lllll View Post
    Show that if $\displaystyle \{a_n\}$ and $\displaystyle \{b_n\}$ are Cauchy in $\displaystyle \mathbb{Q}$ contained in equivalence classes $\displaystyle a$ and $\displaystyle b$ (real numbers) respectively, and if $\displaystyle a_n < b_n$ for all $\displaystyle n$, then there exists a real number $\displaystyle \delta$ which contains a rational Cauchy sequence $\displaystyle \{\delta_n\}$ each of whose $\displaystyle \delta_n$ is positive and $\displaystyle b=a+\delta$.

    Here's what I have, for some reason it doesn't seem correct.

    since $\displaystyle a_n < b_n$ then $\displaystyle 0 < b_n-a_n$ implying that there exist $\displaystyle \delta_n$ such that $\displaystyle \delta_n = b_n-a_n$ for all $\displaystyle n$
    Be careful here. $\displaystyle a_n$ and $\displaystyle b_n$ are rational numbers of sequences in equivalence classes. You don't want to say that $\displaystyle b_n- a_n$ is an equivalence class. $\displaystyle \{\delta_n\}$ is a sequence of rational numbers and will be in an equivalence class (and so define an equivalence class) if you can prove it is a Cauchy sequence.

    but since $\displaystyle {\delta_n}$ is an equivalence class of a Cauchy sequence then $\displaystyle [\{\delta_n\}] = \delta \in \mathbb{R}$ and the same for $\displaystyle [\{a_n}]=a \in \mathbb{R}$ and $\displaystyle [\{b_n}] = b \in \mathbb{R}$ so then $\displaystyle [\{\delta_n\}] = [\{b_n\}]-[\{a_n\}]\Rightarrow \delta = b-a \Rightarrow a+\delta = b$
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