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*Let A be the set of all positive rationals, p such that p*

To do this we associate with each rational p > 0 the number

q = p - ((p

^{2}< 2 and let B consist of all rationals p such that p^{2}> 2. We shall now show that A contains no largest number, and B...To do this we associate with each rational p > 0 the number

q = p - ((p

^{2}- 2) / (p + 2))Now, I can't see where this is going. We want to pick a q that is

**rational**, and always bigger than p but less than root(2) or always smaller than p but greater than root(2) depending on whether we are considering the set A or B.

Because we want q to be rational we can't pick the number directly in between p and root(2), that is (p + root(2) /2).

I guess my question is, how would one come up with that calculation for q? I can't for the life of me wrap my head around where that formula for q would come from if I was trying to pick q on my own..