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Chapter 1: Problem 6

Show that the following sequences of sets, \(\left\\{C_{k}\right\\}\), are nondecreasing, (1.2.16), then find \(\lim _{k \rightarrow \infty} C_{k}\). (a) \(C_{k}=\\{x: 1 / k \leq x \leq 3-1 / k\\}, k=1,2,3, \ldots\). (b) \(C_{k}=\left\\{(x, y): 1 / k \leq x^{2}+y^{2} \leq 4-1 / k\right\\}, k=1,2,3, \ldots\)

Short Answer

Expert verified
Both sequences of sets \(\{C_k\}\) are nondecreasing. The limit as \(k\) approaches infinity for sequence (a) is given by \(\{x: 0 \leq x \leq 3\}\) and for sequence (b) is given by \(\{(x, y):0 \leq x^{2}+y^{2} \leq 4\}\).

Step by step solution

01

Nondecreasing Proof for Part (a)

For \(C_k = \{x: 1 / k \leq x \leq 3-1 / k\}\), note that as \(k\) increases, \(\frac{1}{k}\) decreases and \(3-\frac{1}{k}\) also decreases. Therefore, every \(C_k\) is a subset of \(C_{k+1}\), so the sequence is nondecreasing.
02

Find the Limit for Part (a)

The limit as \(k\) approaches infinity can be found by noting that as \(k\) approaches infinity, \(1 / k\) approaches 0 and \(3-1 / k\) approaches 3. Therefore, the union over all \(C_k\)'s which would be \(lim _{k \rightarrow \infty} C_{k} = \{x: 0 \leq x \leq 3\}\).
03

Nondecreasing Proof for Part (b)

For \(C_k =\{(x, y): 1 / k \leq x^{2}+y^{2} \leq 4-1 / k\}\), as \(k\) increases, \(1 / k\) decreases and \(4-1/k\) also decreases. Therefore, the intersection of \(x^{2}+y^{2}\) with the decreasing interval also decreases, so every \(C_k\) is a subset of \(C_{k+1}\), and the sequence is nondecreasing.
04

Find the Limit for Part (b)

The limit as \(k\) approaches infinity can be found by noting that as \(k\) approaches infinity, \(1/k\) approaches 0, and \(4-1/k\) approaches 4. Therefore, the union over all \(C_k\)'s which would be \(lim _{k \rightarrow \infty} C_{k} =\{(x, y):0 \leq x^{2}+y^{2} \leq 4\}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit of a Sequence
Understanding the limit of a sequence is foundational in calculus and analysis. It's the value that the elements of a sequence get infinitely close to as the index increases without bounds. Imagine a sequence of numbers getting closer and closer to a single number; that single number is the limit.

In our exercise with the set \(C_k = \{x: 1 / k \leq x \leq 3-1 / k\}\), we see that as \(k\) goes to infinity, the values of \(1/k\) and \(3-1/k\) approach 0 and 3, respectively. Hence, the limit of this sequence is the closed interval \[0,3\]. The concept of sequence limits extends beyond numbers to include sets and functions, yielding profound implications in continuous mathematics.
Set Theory
At the heart of modern mathematics lies set theory, a branch that studies sets or collections of objects. It's the foundation for various mathematical disciplines, including analysis, which deals with sequences and their limits. In our exercise, the sets \(C_k\) are collections of real numbers defined by certain conditions.

A key concept in set theory is the idea of a sequence of sets being nondecreasing, which means every set in the sequence is a subset of the ones following it. This property is crucial when deducing the limit of a sequence of sets, as it guarantees that there is a well-defined union (limit) of all sets in the sequence.
Real Number Intervals
The concept of real number intervals is vital in understanding continuous ranges in mathematics. Intervals can be open, closed, or half-open, and they represent all numbers lying between a set of endpoints. The difference between the types of intervals lies in whether they include their endpoints or not.

For instance, in the sequence \(C_k\) from the exercise, each \(C_k\) represents a closed interval since it includes its endpoints, \(1/k\) and \(3-1/k\). As \(k\) becomes larger, these endpoints converge, and we examine the development of these intervals to understand the limit of the sequence.
Mathematical Proofs
A mathematical proof is a logical argument that establishes the truth of a mathematical statement. Proofs use definitions, axioms, previously established statements, and logical deductive reasoning to arrive at a conclusion. In the context of our exercise, the proof shows that the sequences \(C_k\) are nondecreasing.

The proof consists of demonstrating that for each integer \(k\), the set \(C_k\) is a subset of \(C_{k+1}\), which conveys the nondecreasing nature of the sequence. Through mathematical proof, we can rigorously confirm the behavior of the sequence and its limit, adding certainty and understanding to our mathematical worldview.

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Most popular questions from this chapter

Let us select five cards at random and without replacement from an ordinary deck of playing cards. (a) Find the pmf of \(X\), the number of hearts in the five cards. (b) Determine \(P(X \leq 1)\).

Let \(C_{1}\) and \(C_{2}\) be independent events with \(P\left(C_{1}\right)=0.6\) and \(P\left(C_{2}\right)=0.3\). Compute (a) \(P\left(C_{1} \cap C_{2}\right)\), (b) \(P\left(C_{1} \cup C_{2}\right)\), and (c) \(P\left(C_{1} \cup C_{2}^{c}\right)\).

A mode of the distribution of a random variable \(X\) is a value of \(x\) that maximizes the pdf or pmf. If there is only one such \(x\), it is called the mode of the distribution. Find the mode of each of the following distributions: (a) \(p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere. (b) \(f(x)=12 x^{2}(1-x), 0

In a certain factory, machines I, II, and III are all producing springs of the same length. Machines I, II, and III produce \(1 \%, 4 \%\), and \(2 \%\) defective springs, respectively. Of the total production of springs in the factory, Machine I produces \(30 \%\), Machine II produces \(25 \%\), and Machine III produces \(45 \%\). (a) If one spring is selected at random from the total springs produced in a given day, determine the probability that it is defective. (b) Given that the selected spring is defective, find the conditional probability that it was produced by Machine II.

Let \(X\) be the number of gallons of ice cream that is requested at a certain store on a hot summer day. Assume that \(f(x)=12 x(1000-x)^{2} / 10^{12}, 0
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