let A be a set of all positive rational number such that [itex]p^2<2[/itex]
B be a set of all positive rational number such that [itex]p^2>2[/itex]
The Attempt at a Solution
Set A is clearly non empty, and is a subset of real number, anyway i can choose 3 is upperbound, therefore upperbound exist, so by completeness axiom, supremum exist.
But the book here said
"Set A is bounded above, in fact every element in B a the upperbound of A. Since B has no smallest element, A has no least upper bound/ supremum in Q."
i'm really sure i'm not wrong. But am i wrong?
p/s; i just realised that this book define least-upper-bound property(more general case from completeness axiom), and also above example are the counterexample that proves Q does not have least-upper-bound property(follows from what the book have shown, not mine).
But aren't this contradicting the completeness axiom?
since Q is a subset of R, and any non-empty subset of Q that bounded from above has supremum(from completeness axiom), therefore Q has the least-upper-bound property.
help, where i gone wrong T_T