Question

Prove that $\cap_{\alpha \in \mathbf{R}_{+}}[-\alpha, \alpha]=\{0\}$.

   Prove that $\cap_{\alpha \in \mathbf{R}_{+}}[-\alpha, \alpha]=\{0\}$.

Competitive Equilibrium: Theory and Applications

Competitive Equilibrium: Theory and Applications

Bryan Ellickson 1st Edition

Chapter 1, Problem 2

Instant Answer

Solved by Expert Tatiana Graham

06/05/2021

Step 1

This means that $x$ is in the interval $[-\alpha, \alpha]$ for every positive real number $\alpha$. Since $x$ is in the interval $[-\alpha, \alpha]$ for every positive real number $\alpha$, it must be the case that $-\alpha \leq x \leq \alpha$ for every positive Show more…

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Prove that $\cap_{\alpha \in \mathbf{R}_{+}}[-\alpha, \alpha]=\{0\}$.

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Key Concepts

Intersection of Sets

This concept refers to the set that contains all elements that are common to every set in a given collection. In set theory, an element belongs to the intersection if and only if it is a member of each individual set being intersected. It is a fundamental operation used to gather shared attributes or commonalities across multiple sets.

Nested Intervals

Nested intervals are a sequence or collection of intervals where each interval is contained within the previous one. In analysis, this property often allows one to deduce information about the limit set or the common points of all these intervals, leveraging the fact that as the intervals become arbitrarily small, their intersection tends to a minimal or unique value.

Absolute Value and Symmetry in Real Numbers

The absolute value function measures the distance of a real number from zero, regardless of its sign, and is particularly important when dealing with symmetric intervals centered at zero. Understanding how the absolute value relates to the bounds of these intervals helps in proving that if an element has a nonzero absolute value, it will eventually fall outside some sufficiently small symmetric interval, thereby isolating zero as the only element common to all such intervals.

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