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.
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.