In showing that $\displaystyle \mathbb{Q} $ does not follow the least upper bound property consider the following sets: $\displaystyle A = \{p: p^2 <2, \ p \in \mathbb{Q} \} $ and $\displaystyle B = \{p: p^2 >2, \ p \in \mathbb{Q} \} $. We want to show that for every $\displaystyle p \in A $ we can find a $\displaystyle q \in A $ such that $\displaystyle p<q $. Likewise, for every $\displaystyle p \in B $, we can find a $\displaystyle q \in B $ such that $\displaystyle q < p $. To do this, we let:

$\displaystyle q = p-\frac{p^2-2}{p+2} $ (1)

and it all works out (e.g. (i) $\displaystyle p \in A \Rightarrow p<q $ and (ii) $\displaystyle p \in B \Rightarrow q <p $). We also consider $\displaystyle q^2-2 $ to show that $\displaystyle q $ is also in $\displaystyle A $ or $\displaystyle B $ respectively.

The question is, how do we come up with (1)? Was it just a guess and check process? It seems like the second term in (1) is very small. So in a sense we are considering $\displaystyle q= p- \varepsilon $. But how do we come up with $\displaystyle \varepsilon $? Is it arbitrary?