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October 10th, 2010, 02:40 PM   #1
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dedekind cut

show that {r IN Q: r^3 < 2} is a dedekind cut.

{r in Q: r^3 < 2} = {r in Q: r <(2^(1/3))}

i) ? is not in Q and ? is nonempty.
? is nonempty and 2^(1/3) is not in Q so ? is not in Q.

ii) if r in ?, s in Q and s<r then s in ?
suppose r in ?, s in Q and s<r. so s<r<2^(1/3). Hence s < 2^(1/3) therefore s in ?.

iii) ? contains the no largest rational.
by contradiction. Let p be the largest ?. Then p < 2^(1/3). By denseness of Q there exists q in Q s.t p<q<2^(1/3). This contradiction since p is the largest in ?. so q in not in ?.
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