Problem 128

Question

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.

Step-by-Step Solution

Verified

Answer

Assuming different least upper bounds leads to a contradiction, thus $M_1 = M_2$. An upper bound must be unique.

1Step 1: Understanding Least Upper Bounds

A least upper bound (lub) for a set, also known as the supremum, is the smallest number that is greater than or equal to every element in the set. In this problem, we have $M_1$ and $M_2$ as least upper bounds for the sequence $\{a_n\}$. This means both $M_1$ and $M_2$ are the smallest numbers bigger than all elements in the sequence.

2Step 2: Assume Two Least Upper Bounds

Let's start with the assumption that $M_1$ and $M_2$ are both least upper bounds for $\{a_n\}$ but $M_1 eq M_2$. We aim to show this assumption leads to a contradiction, thereby proving that $M_1 = M_2$.

3Step 3: Compare the Bounds

Assume without loss of generality that $M_1 < M_2$. Given that $M_2$ is an upper bound, this implies $M_2$ is greater than or equal to every element in the sequence. However, since $M_1$ is the least upper bound, no number less than $M_1$ could be an upper bound.

4Step 4: Deriving a Contradiction

Since $M_1 < M_2$, there must exist an element in $\{a_n\}$ that is greater than $M_1$ but less than or equal to $M_2$. However, this contradicts the fact that $M_1$ is a least upper bound. Accordingly, it must be that both $M_1$ and $M_2$ are equal due to their minimality as upper bounds.

Key Concepts

SupremumSet TheoryMathematical Proofs

Supremum

The concept of the supremum is integral to understanding the behavior of sets in mathematics, particularly in real analysis. The supremum (also referred to as the least upper bound) of a set is essentially the smallest real number that is greater than or equal to every element in that set. This definition aligns with its Latin origin 'supremus', meaning 'highest'.

Finding the supremum is crucial not only in defining boundaries for sequences but also in ensuring that no number smaller than this value can serve as a comprehensive upper bound. For example, consider a sequence $a_n$ that approaches, but never exceeds, a certain limit. The supremum of $a_n$ would be this limit itself, such that no other number below it encompasses all elements within the sequence.

The supremum may or may not be an element of the set. If it is part of the set, it is called the maximum.
If no such upper bound smaller than the given supremum exists, we can definitively say that it is indeed the least upper bound.

In practical terms, proving that two seeming supremum candidates (like $M_1$ and $M_2$) are equal, directly supports their uniqueness in any given context, ensuring the integrity of mathematical boundaries within set theory.

Set Theory

Set theory serves as the foundational bedrock of modern mathematics, offering us a way to conceptualize collections of objects. This discipline aids in understanding not just numbers and functions but frameworks for numerous mathematical constructs. In the context of sequences and bounds, set theory plays a vital role.

Sets are used to define sequences, like the collection of terms $\{a_n\}$. Within this collection, set theorists explore various properties, especially bounding properties like least upper bounds (suprema).

A sequence is essentially an ordered set, often infinite, comprising elements that potentially escalate towards a limit.
In proving properties like the uniqueness of the supremum, it becomes paramount to understand the order and arrangement of these elements.

Set theory also identifies relationships between elements, facilitating comparisons within sequences that underpin proofs of concepts such as the uniqueness of the supremum. The language and principles of set theory extend into greater mathematical proofs, allowing for a systematic approach to problem solving.

Mathematical Proofs

Mathematical proofs are the cornerstone of validating and affirming truths within mathematics. They offer a structured argument that establishes the veracity of a certain claim based on known axioms, definitions, and previously established theorems.

When tackling a proof, like demonstrating the uniqueness of least upper bounds, the process begins by assuming a statement is opposite to what we wish to prove, known as 'proof by contradiction'. Let's see how this applies efficiently:

**Initial Assumption**: Assume two different least upper bounds, $M_1$ and $M_2$, exist for sequence $\{a_n\}$.
**Logical Deduction**: Suppose $M_1 < M_2$, yet show that no sequence element exceeds $M_1$, contradicting its supposed supremum nature.
**Conclusion**: Deduce that since such arrangement leads to a contradiction, the initial assumption must be false, concluding $M_1 = M_2$.

These proofs showcase not only the power of logic but also illustrate how rigorous reasoning can resolve apparent complexities into simplified truths. It is through these techniques that mathematicians ensure their findings are reliable, precise, and applicable across various scenarios within and beyond mathematical confines.

Problem 127

Problem 130

Other exercises in this chapter

Problem 126

In Exercises $125-126,$ use the definition of convergence to prove the given limit. $$\lim _{n \rightarrow \infty}\left(1-\frac{1}{n^{2}}\right)=1$$

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Problem 127

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Problem 130

Prove that if $\left\\{a_{n}\right\\}$ is a convergent sequence, then to every positive number $\epsilon$ there corresponds an integer $N$ such that for a

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Problem 131

Uniqueness of limits Prove that limits of sequences are unique. That is, show that if $L_{1}$ and $L_{2}$ are numbers such that $a_{n} \rightarrow L_{1}$

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