I am reading this paragraph in little Rudin, right at the beginning. Let A be the set of all positive rationals, p such that p2 < 2 and let B consist of all rationals p such that p2 > 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 - ((p2 - 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..