Let A be a bounded subset of ℝ containing at least two points. Prove: (a) -∞<infA<supA<+∞. (b) I

Step 1: Prove part (a) $u002Du005Cinftyu003Cu005Cinf Au003Cu005Csup Au003C+u005Cinfty$.u000A u002D Since $A$ is bounded, there exist r

Let A be a bounded subset of ℝ containing at least two...

Let $A$ be a bounded subset of $\mathbb{R}$ containing at least two points. Prove: (a) $-\infty<\inf A<\sup A<+\infty$. (b) If $B$ is a nonempty subset of $A$, then $\inf A \leq \inf B \leq \sup B \leq \sup A$. (c) If $B$ is the set of all upper bounds for $A$, then $B$ is nonempty, bounded below, and $\inf B=\sup A$.

Let $A$ be a bounded subset of $\mathbb{R}$ containing at least two points. Prove:
(a) $-\infty<\inf A<\sup A<+\infty$.
(b) If $B$ is a nonempty subset of $A$, then $\inf A \leq \inf B \leq \sup B \leq \sup A$.
(c) If $B$ is the set of all upper bounds for $A$, then $B$ is nonempty, bounded below, and $\inf B=\sup A$.

Key Concepts

Bounded Set

A bounded set in the real numbers is a set that is contained within some finite interval; that is, there exist real numbers m and M such that every element of the set lies between m and M. This concept is vital because many of the properties like the existence of supremum and infimum depend on the set being bounded.

Supremum

The supremum (or least upper bound) of a set is the smallest number that is greater than or equal to every element in the set. In the context of bounded sets in ?, the completeness property of the real numbers ensures that every nonempty bounded-above set has a supremum.

Infimum

The infimum (or greatest lower bound) of a set is the largest number that is less than or equal to every element in the set. Similar to the supremum, every nonempty bounded-below set in ? is guaranteed to have an infimum due to the completeness property of the real numbers.

Completeness of ?

This property states that every nonempty subset of the real numbers that is bounded above has a least upper bound (supremum), and every nonempty subset bounded below has a greatest lower bound (infimum). This is a foundational concept in real analysis that underpins many properties of real number sets.

Order-Preserving Properties of Bounds

This concept involves understanding how the extremal values (infimum and supremum) of a set relate to those of its subsets. If one set is a subset of another, then the supremum and infimum of the subset will lie within the bounds of the original set, reflecting the order preserving nature of these operations.

Set of Upper Bounds

For a given set in ?, the set of all upper bounds is the collection of numbers that are greater than or equal to every element in the set. This set is itself nonempty and bounded below, and its infimum corresponds to the supremum of the original set. This link is a direct consequence of the definitions of supremum and the completeness property.

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