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Chapter 9: Problem 128

Uniqueness of least upper bounds Show that if \(M_{1}\) and \(M_{2}\) are least upper bounds for the sequence \(\left\\{a_{n}\right\\},\) then \(M_{1}=M_{2}\) That is, a sequence cannot have two different least upper bounds.

Short Answer

Expert verified
The least upper bound of a sequence is unique, so \(M_1 = M_2\).

Step by step solution

01

Define Least Upper Bound

A least upper bound or supremum of a sequence \(\{a_n\}\) is a number \(M\) such that every element of the sequence is less than or equal to \(M\) and for any number \(m\) less than \(M\), there exists an element of the sequence that is greater than \(m\). Thus, \(M\) is the smallest number that an sequence can be less than or equal to.
02

Assume Two Least Upper Bounds

Assume that \(M_1\) and \(M_2\) are both least upper bounds (suprema) for the sequence \(\{a_n\}\). This means both \(M_1\) and \(M_2\) satisfy the condition of being the smallest number that bounds the sequence from above.
03

Compare Least Upper Bounds

If \(M_1\) and \(M_2\) are both least upper bounds, let's compare them. Without loss of generality, assume \(M_1 \leq M_2\). Due to the definition of a least upper bound, we also have \(M_2 \leq M_1\) because \(M_2\) is also the least upper bound.
04

Conclude that Least Upper Bounds are Equal

From the assumptions \(M_1 \leq M_2\) and \(M_2 \leq M_1\), it follows that \(M_1 = M_2\). This shows that the least upper bound is unique: if there are two least upper bounds, they must be equal.

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Key Concepts

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

Understanding Supremum
The concept of supremum, also known as the least upper bound, is fundamental in analyzing the behavior of sequences. When we talk about the supremum of a sequence \(*a_n*\), we are looking at the smallest number that the entire sequence does not exceed. Essentially, this means that for any given element within the sequence, it will always be less than or equal to the supremum.

A crucial property is that if you take any number less than this supremum, there must be an element in the sequence that is greater than this number. This ensures that the supremum is not an arbitrary number but indeed the smallest such number that functions as an upper boundary. Supremum is particularly crucial in cases where the sequence does not have an actual maximum value within its set but does have a boundary that it does not surpass.
Exploring Sequences
Sequences are ordered collections of numbers, often following a specific rule or pattern. Understanding sequences is pivotal since many mathematical concepts, including the supremum, hinge on their properties.

Commonly, sequences are written in the form \(*a_1, a_2, a_3, \ldots*\) where *a_n* refers to the *n*-th term. Analyzing sequences helps in making informed deductions about their limits and bounds.

Depending on the rule, sequences can be finite or infinite and may converge or diverge. A convergent sequence approaches a particular value as *n* increases, whereas a divergent sequence does not settle to a single value.

In many cases, the importance of sequences lies in how they interact with their boundaries, such as their least upper bounds, when evaluating their overall behavior and convergence properties.
Defining Upper Bounds
Upper bounds serve as a threshold that elements of a sequence cannot exceed. If a number \(*M*\) is an upper bound for a sequence \(*a_n*\), every element of the sequence satisfies the condition \(*a_n \leq M*\).

Identifying upper bounds is crucial as they help describe and constrain sequences, providing a framework within which the sequence exists.

The least upper bound, or supremum, is the smallest possible of these bounds. It鈥檚 worth noting that while a sequence may have multiple upper bounds, the least upper bound is always unique. This aspect is integral to understanding that even if a sequence is unbounded or approaches infinity, its behavior is still manageable through finding upper bounds.

In mathematical analysis, determining upper bounds and understanding their properties enhances the ability to analyze limits and continuity, grounding more complex mathematical theories.

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Most popular questions from this chapter

Any real number in the interval [0,1] can be represented by a decimal (not necessarily unique) as $$ 0 . d_{1} d_{2} d_{3} d_{4} \ldots=\frac{d_{1}}{10}+\frac{d_{2}}{10^{2}}+\frac{d_{3}}{10^{3}}+\frac{d_{4}}{10^{4}}+\cdots $$ where \(d_{i}\) is one of the integers \(0,1,2,3, \ldots,\) 9. Prove that the series on the right-hand side always converges.

Determine if the sequence is monotonic and if it is bounded. $$a_{n}=\frac{3 n+1}{n+1}$$

Show that neither the Ratio Test nor the Root Test provides information about the convergence of $$ \sum_{n=2}^{\infty} \frac{1}{(\ln n)^{p}} \quad(p \text { constant }) $$

Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: a. For what values of \(x\) can the function be replaced by each approximation with an error less than \(10^{-2} ?\). b. What is the maximum error we could expect if we replace the function by each approximation over the specified interval? Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals. Step \(I:\) Plot the function over the specified interval. Step 2: Find the Taylor polynomials \(P_{1}(x), P_{2}(x),\) and \(P_{3}(x)\) at \(x=0\) Step 3: Calculate the \((n+1)\) st derivative \(f^{(n+1)}(c)\) associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of \(c\) over the specified interval and estimate its maximum absolute value, \(M .\) Step 4: Calculate the remainder \(R_{n}(x)\) for each polynomial. Using the estimate \(M\) from Step 3 in place of \(f^{(n+1)}(c),\) plot \(R_{n}(x)\) over the specified interval. Then estimate the values of \(x\) that answer question (a). Step 5: Compare your estimated error with the actual error \(E_{n}(x)=\left|f(x)-P_{n}(x)\right|\) by plotting \(E_{n}(x)\) over the specified interval. This will help answer question (b). Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5. $$f(x)=(1+x)^{3 / 2}, \quad-\frac{1}{2} \leq x \leq 2$$

Find the first four nonzero terms in the Maclaurin series for the functions. $$\sin \left(\tan ^{-1} x\right)$$
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