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Chapter 2: Problem 7

Let \(I_{n}:=[0,1 / n]\) for \(n \in \mathbb{N}\). Prove that \(\bigcap_{n=1}^{\infty} I_{n}=\\{0\\}\).

Short Answer

Expert verified
The intersection of the infinite sets \(I_n\) given by \(\bigcap_{n=1}^{\infty} I_{n} \{0\}\) only includes the single point '0'.

Step by step solution

01

Understand the series of sets

The given series of sets \(I_n\) are intervals starting from 0 to 1/n. As n increases, the interval [0, 1/n] becomes smaller.
02

Analyze the intersection

The intersection of the sets \(I_n\) represents all elements that are common to all \(I_n\). Since each \(I_n\) starts at 0, all intersections will always include 0. As n becomes larger \(1/n\) become increasingly smaller. So, intersection of all the intervals is a single point [0].
03

Mathematical proof

Reflecting on the previous steps, we can deduct that for any \(c>0\), there exists an \(n \in \mathbb{N}\) such that \(1/n < c\). Thus, c cannot be a part of the intersection of all \(I_n\). Because if c is part of the intersection, it must be part of every \(I_n\) which is not the case for sufficiently large n. Therefore, the only possibility left is that the intersection is {0}. This satisfies the definition of an intersection of a series of sets and proves the exercise statement.

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Most popular questions from this chapter

(a) If \(0

Let \(X\) be a nonempty set, and let \(f\) and \(g\) be defined on \(X\) and have bounded ranges in \(\mathbb{R}\). Show that $$ \sup \\{f(x)+g(x): x \in X\\} \leq \sup (f(x): x \in X\\}+\sup \\{g(x): x \in X\\} $$ and that $$ \inf \\{f(x): x \in X\\}+\inf \\{g(x): x \in X\\} \leq \inf \\{f(x)+g(x): x \in X\\} $$ Give examples to show that each of these inequalities can be either equalities or strict inequalities.

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