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Show D= {x: x [tex]\in[/tex] Q and (x [tex]\leq[/tex] or x^2 < 2)} is a dedekind cut.

A set D c Q is a Dedekind set if

1)D is not {}, D is not Q

2) if r[tex]\in[/tex] D then there exists a s [tex]\in[/tex] D s.t r<s

3) if r [tex]\in[/tex] D and if s [tex]\leq[/tex] r, then s [tex]\in[/tex] D.

For the first case, D is not an empty set because x is equal to 0 or the sqrt of 2. But, how do I prove case 2,3. Do I have to use addition/multiplication to prove them?

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# Dedekind Cuts

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