October 10th, 2010, 02:40 PM  #1 
Senior Member Joined: Nov 2009 Posts: 129 Thanks: 0  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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