How do I show that A={r in Q: r^3<2} is a Dedekind cut.(adsbygoogle = window.adsbygoogle || []).push({});

Here is the definition I am working with.

A subset A of Q is a Dedekind cut if and only if A satisfies the

following 3 properties:

(i) A is a proper nonempty subset of Q.

(ii) If r is in A, s in Q, and s<r, then s is in A.

(iii) A contains no greatest rational.

I showed that A satisfies (i) and (ii). I noticed that 5/4 is in A

and I tried to find a rational greater than 5/4 whose cube is less

than 2. I looked at the sequence (5n+1)/4n, and I found that n=26

works. That is, (4/5)<(131/104) and (131/104)^3 < 2.

So I think that if a/b is any rational with b>0 and (a/b)^3 < 2, then

there should be some positive integer n such that [(an+1)/bn]^3 < 2. But I don't know

how to show that this n exists.

I tried contradiction:

Let a/b be in A with b>0 , and assume that [(an+1)/bn]^3 >= 2 for all

positive n. Then (a/b)^3 < 2 <= [(an+1)/bn]^3 for all positive n. I

think this may imply that 2^(1/3) is rational, which I know is not true.

Am I on the right track?

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# Dedekind cut

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