Denseness and Completeness of Real Numbers

The property of denseness states that between any two different real numbers there is always:

True or False: The property of denseness holds for the set of integers.

A set A is bounded above if there exists a real number u such that:

A number M is the least upper bound (l.u.b.) of a set A if M is an upper bound and:

Classify each set as Bounded or Unbounded

Classify each set as Bounded or Unbounded

For the set \(A = \{1, \tfrac12, \tfrac13, \tfrac14, \ldots\}\), what is the least upper bound (l.u.b.)?

For the same set \(A = \{1, \tfrac12, \tfrac13, \tfrac14, \ldots\}\), what is the greatest lower bound (g.l.b.)?

The Axiom of Least Upper Bound states that every nonempty subset of R with an upper bound has:

Does the least upper bound axiom hold if R is replaced by Q, the set of rational numbers?

Consider \(A = \{x \in \mathbb{Q} : x > 0 \text{ and } x^2 < 2\}\). This set is bounded above in Q but has no least upper bound within Q. What is its least upper bound in R?

Consider \(A = \{x \in \mathbb{Q} : x > 0 \text{ and } x^2 < 2\}\). This set is bounded above in Q but has no least upper bound within Q. What is its least upper bound in R?