In showing that does not follow the least upper bound property consider the following sets: and . We want to show that for every we can find a such that . Likewise, for every , we can find a such that . To do this, we let:
(1)
and it all works out (e.g. (i) and (ii) ). We also consider to show that is also in or 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 . But how do we come up with ? Is it arbitrary?