Question

What is the least upper bound property for nonempty subsets of real numbers? Does the least upper bound property hold for subsets of the rational numbers? Does it hold for subsets of the integers?

Short answer

Consider Sis a nonempty subset of real numbers and M is a real number such that the element of the subset $x\in S\&x\le M$then the M is said to be the least upper bound.

The Least upper bound property does not hold for subsets of the rational numbers.

The Least upper bound property hold for subsets of the integers.

Step-by-step solution

Step 1. Given information. 

The given topics that need to be explained are the followings.

• least upper bound property for nonempty subsets of real numbers.
• least upper bound property hold for subsets of the rational numbers.
• least upper bound property hold for subsets of the integers.

Step 2. least upper bound property for nonempty subsets of real numbers.

Least upper bound property for nonempty subsets of real numbers.

Consider Sis a nonempty subset of real numbers and M is a real number such that the element of the subset $x\in S\&x\le M$then the M is said to be the least upper bound.

Step 3. least upper bound property for subsets of the rational numbers.

The Least upper bound property does not hold for subsets of the rational numbers.

Consider $S=\left\{x:x\in Q\&x<5\right\}$

Here nonempty subset S of rational numbers $\left(S\in Q\right)$has an upper bound role="math" localid="1649182930779" $7$but subset S does not hold the least upper bound property.

Step 4. least upper bound property for subsets of the integers.

The Least upper bound property hold for subsets of the integers.

Consider $S=\left\{x:x\in Z\&x\le 7\right\}$

Here nonempty subset Sof rational numbers $\left(S\in Z\right)$has an upper bound $7$and subset S holds the least upper bound property.