Greatest lower bound/least upper bound in Q

In summary, the conversation is discussing the boundedness of a set A containing rational numbers, where the condition is that n is not a perfect square. It is mentioned that A is bounded in \mathbb{Q}, but does not have a greatest lower bound or a least upper bound in \mathbb{Q}. The speaker suggests using the proof of the irrationality of √n, when n is not a perfect square, to prove this.
  • #1
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I have the following question:

Let [itex]n\in\mathbb{Z}^{+}[/itex] st. [itex]n[/itex] is not a perfect square. Let [itex]A=\{x\in\mathbb{Q}|x^{2}<n\}[/itex]. Show that [itex]A[/itex] is bounded in [itex]\mathbb{Q}[/itex] but has neither a greatest lower bound or a least upper bound in [itex]\mathbb{Q}[/itex].

To show that [itex]A[/itex] is bounded in [itex]\mathbb{Q}[/itex] I have to show that it has a infimum and a supremum in [itex]\mathbb{Q}[/itex], right? Not sure where to start...
 
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  • #2
No, only thing you need to show boundedness is that there are rationals q,q' , such that

q2<n and n<q'2. For the rest, I'm not sure what material you're familiar with. If you know the proof of the irrationality of √2 , you can show that this extends to the irrationality of √n , when n is not a perfect square.
 

What is the greatest lower bound in Q?

The greatest lower bound in Q, also known as the infimum, is the largest number that is less than or equal to all elements in a given set of rational numbers Q. It is denoted as inf(Q) or glb(Q).

What is the least upper bound in Q?

The least upper bound in Q, also known as the supremum, is the smallest number that is greater than or equal to all elements in a given set of rational numbers Q. It is denoted as sup(Q) or lub(Q).

Why are the greatest lower bound and least upper bound important in Q?

The greatest lower bound and least upper bound provide a way to define limits and continuity in the set of rational numbers Q. They also help in determining if a set of rational numbers is bounded or not.

How are the greatest lower bound and least upper bound calculated in Q?

The greatest lower bound and least upper bound can be calculated by finding the minimum and maximum values in a given set of rational numbers Q, respectively. Alternatively, they can also be calculated using the limit of a sequence of rational numbers.

Can the greatest lower bound and least upper bound exist in Q if the set is unbounded?

No, the greatest lower bound and least upper bound in Q can only exist if the given set of rational numbers is bounded. If the set is unbounded, then the greatest lower bound and least upper bound are said to be infinite or do not exist.

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