If A is a nonempty subset of ℝ that is bounded below, show that A has a greatest lower bound. Th

Step 1: Define the set u005C(u002DAu005C).u000ASince u005C(Au005C) is a nonempty subset of u005C(u005Cmathbb{R}u005C) and is bounded

If A is a nonempty subset of ℝ that is bounded below, show...

If $A$ is a nonempty subset of $\mathbb{R}$ that is bounded below, show that $A$ has a greatest lower bound. That is, show that there is a number $m \in \mathbb{R}$ satisfying: (i) $m$ is a lower bound for $A$; and (ii) if $x$ is a lower bound for $A$, then $x \leq m$. [Hint: Consider the set $-A=\{-a: a \in A\}$ and show that $m=-\sup (-A)$ works.]

Key Concepts

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. It is a fundamental concept in real analysis that provides a way to describe the lower limit of bounded sets, ensuring that even if an actual minimum does not exist in the set, there is still a well-defined lower bound that is as tight as possible.

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. This concept is closely related to the infimum, serving as the upper analogue. Supremum is crucial in dealing with bounded sets and is used to formally handle limits and bounds in mathematical analysis.

Completeness Axiom

The completeness axiom of the real numbers states that every nonempty subset of the real numbers that is bounded above has a supremum (least upper bound). This property is essential in real analysis as it ensures the existence of limits and bounds needed for rigorous definitions of continuity, convergence, and many other analytical concepts.

Set Transformation

Transforming a set by applying a function, such as negation, is a useful technique in analysis. In the context of this problem, considering the set -A, where each element of A is negated, leverages the relationship between suprema and infima. This duality shows that finding a supremum in the transformed set corresponds to finding an infimum in the original set, simplifying the proof of the existence of greatest lower bounds.

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