Conclusion : This question makes no sense unless we consider it in the set of real numbers
How to prove the greatest lower bound for all positive rational numbers is $0$?
I can only figure out the following right now:
(1). $0$ is a lower bound for all positive rational numbers;
(2). Any positive rational number is not the greatest lower bound for all positive rational numbers;
Density property of rational numbers may be helpful to prove the conclusion, but it is not easy for me to give a complete rigorous proof right now.
I think
1). There is no guarantee that the greatest lower bound in the question must be a rational number .
2).The question doesn't want to prove that zero is the greatest lower bound in Q, I don't need the proof that 0 is the greatest rational lower bound.
Best Answer
Argue by contradiction. Suppose $c\in\Bbb Q$ satisfies $c>0$ and $c$ is a lower bound for all rational numbers. Then $c/2\in\Bbb Q$ and $c/2<c$ since if $c/2\geq c$ then $1/2\geq1$. A contradiction.
Thus the greatest lower bound must be less or equal to zero. Now zero is the largest number in that range and zero works so it must be the greatest lower bound.