Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 63
Prove that, if \(\left\\{a_{n}\right\\}_{n=1}^{\infty}\) converges to a real number \(\ell\), then \(\lim _{n \rightarrow \infty}\left(a_{n}-a_{n+1}\right)=0 .\)
Short Answer
Expert verified
If the sequence converges to \( \ell \), the difference \( (a_n - a_{n+1}) \) also approaches 0 as \( n \to \infty \).
Step by step solution
01
Understand the Problem
We are given a sequence \( \{a_n\}_{n=1}^{\infty} \) that converges to a real number \( \ell \). We need to prove that the limit of the difference between consecutive terms, \( a_n - a_{n+1} \), as \( n \) approaches infinity is 0.
02
Apply Definition of Convergence
By the definition of convergence, since \( \{a_n\} \) converges to \( \ell \), for every \( \epsilon > 0 \), there exists a natural number \( N \) such that for all \( n \geq N \), \( |a_n - \ell| < \epsilon/2 \).
03
Express \( a_n - a_{n+1} \) Using \( \ell \)
Note that \( a_n - a_{n+1} = (a_n - \ell) - (a_{n+1} - \ell) \). We need to show this expression approaches zero as \( n \to \infty \).
04
Apply Triangle Inequality
Using the triangle inequality, we have \(|a_n - a_{n+1}| = |(a_n - \ell) - (a_{n+1} - \ell)| \leq |a_n - \ell| + |a_{n+1} - \ell|\).
05
Bound Each Term
For \( n \geq N \), we know \(|a_n - \ell| < \epsilon/2\) and \(|a_{n+1} - \ell| < \epsilon/2\), thus \(|a_n - a_{n+1}| \leq \epsilon/2 + \epsilon/2 = \epsilon\).
06
Conclusion on Limit
Since \( |a_n - a_{n+1}| < \epsilon \) for all \( n \geq N \) and for any \( \epsilon > 0 \), it follows that \( \lim_{n \to \infty} (a_n - a_{n+1}) = 0 \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence of Sequences
When we talk about the convergence of a sequence, we are referring to how the sequence behaves as it extends to infinity. Specifically, a sequence \( \{a_n\} \) is said to converge to a real number \( \ell \) if the terms of the sequence get arbitrarily close to \( \ell \) as \( n \) becomes very large.
Essentially, this means that for any small positive number \( \epsilon \), however tiny, there's always a point in the sequence beyond which every term is within \( \epsilon \) distance from the limit \( \ell \).
This concept helps us understand how sequences "settle down" into a steady state. Only sequences with such behavior are considered convergent. It provides a powerful way to predict the long-term behavior of sequences and functions, which is a foundational idea in calculus and real analysis.
Essentially, this means that for any small positive number \( \epsilon \), however tiny, there's always a point in the sequence beyond which every term is within \( \epsilon \) distance from the limit \( \ell \).
This concept helps us understand how sequences "settle down" into a steady state. Only sequences with such behavior are considered convergent. It provides a powerful way to predict the long-term behavior of sequences and functions, which is a foundational idea in calculus and real analysis.
Limit Definition
The limit definition is a formal way to describe the behavior of sequences and functions as they approach some point. Specifically for sequences, the limit \( \lim_{n \to \infty} a_n = \ell \) means that as \( n \) approaches infinity, the terms \( a_n \) get closer and closer to the value \( \ell \).
The definition involves the notion of \( \epsilon \) which stands for an arbitrary small positive number. For every \( \epsilon > 0 \), there exists a natural number \( N \) such that for all \( n \geq N \), the distance between \( a_n \) and \( \ell \) is less than \( \epsilon \).
This formal mathematical definition allows us to rigorously prove that a sequence has a limit. It removes the ambiguity and gives us a clear rule to check the convergence of any sequence towards its limit.
The definition involves the notion of \( \epsilon \) which stands for an arbitrary small positive number. For every \( \epsilon > 0 \), there exists a natural number \( N \) such that for all \( n \geq N \), the distance between \( a_n \) and \( \ell \) is less than \( \epsilon \).
This formal mathematical definition allows us to rigorously prove that a sequence has a limit. It removes the ambiguity and gives us a clear rule to check the convergence of any sequence towards its limit.
Triangle Inequality
The triangle inequality is a fundamental concept used frequently in mathematics to establish bounds and relationships between numbers. It states that for any real numbers \( a \) and \( b \), we have:
In our exercise, it sets up an approach to bound the expression \( |a_n - a_{n+1}| \) by relating it to \( |a_n - \ell| \) and \( |a_{n+1} - \ell| \).
Such inequalities help simplify and solve complex problems by breaking them into manageable chunks that can be individually bounded and controlled.
- \( |a + b| \leq |a| + |b| \)
- \( |a - b| \leq |a| + |b| \)
In our exercise, it sets up an approach to bound the expression \( |a_n - a_{n+1}| \) by relating it to \( |a_n - \ell| \) and \( |a_{n+1} - \ell| \).
Such inequalities help simplify and solve complex problems by breaking them into manageable chunks that can be individually bounded and controlled.
Real Analysis
Real analysis is a branch of mathematical analysis dealing with real numbers and real-valued sequences and functions. It lays the groundwork for understanding continuous functions, differentiation, integration, and more complex mathematical concepts.
A major goal of real analysis is to systematically explore limits, continuity, and how sequences and functions behave generally. The rigorous methods and logical arguments in real analysis help ensure that our conclusions about real numbers and functions are both precise and reliable.
In the current exercise, real analysis provides tools and definitions, such as the concept of limits and the technique of using the triangle inequality, to rigorously show how the sequence \( \{a_n - a_{n+1}\} \) behaves as \( n \) goes to infinity. It enables us to handle small differences reliably and achieve a better understanding of the nature of sequences and their limits.
A major goal of real analysis is to systematically explore limits, continuity, and how sequences and functions behave generally. The rigorous methods and logical arguments in real analysis help ensure that our conclusions about real numbers and functions are both precise and reliable.
In the current exercise, real analysis provides tools and definitions, such as the concept of limits and the technique of using the triangle inequality, to rigorously show how the sequence \( \{a_n - a_{n+1}\} \) behaves as \( n \) goes to infinity. It enables us to handle small differences reliably and achieve a better understanding of the nature of sequences and their limits.
