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Let a (Dedekind) cut $r=\{p \in \mathbb{Q} :p^2<2 \text{ or } p<0\}$ and a cut $2^*=\{t\in \mathbb{Q} : t<2\}$. I want to prove $r^2=2^*$. I could show that $r^2 \subset 2^*$ easily, but I couldn't show that $2^* \subset r^2$. How to show that there is $p$, $p' \in r$ such that $t \leq pp' <2$ or $t \leq p^2<2$?

  • 1
    How have you defined multiplication of Dedekind cuts?2012-11-13
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    The definition is followed by rudin's textbook, PMA.
    for cuts a ,b >0, ab={p:p<=rs, for some r $\in$ a and s $\in$ b, r>0, s>0}
    2012-11-13
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    First one is easy just make $p'=1$ ;)2013-01-23

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