Problem 4 For any $b \in \mathbb{R}$, pr... [FREE SOLUTION]

91影视

Introduction to Real Analysis

Robert G. Bartle, Donald R. Sherbert

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 3: Problem 4

For any $b \in \mathbb{R}$, prove that $\lim (b / n)=0$.

Short Answer

Expert verified

The limit $\lim_{n \to \infty} (b / n) = 0$ for any real number $b$.

Step by step solution

Setup the Problem

We are given the limit $\lim_{n \to \infty} (b / n) = 0$, where $b$ is any real number. We are asked to prove this limit. To do so, we need to find a number $N > 0$ that for all $n > N$, the absolute difference of the function $f(n) = b / n$ and the limit 0 is less than any arbitrarily small positive number, $\epsilon$. Therefore, we set up the expression as $|f(n) - 0| < \epsilon$.

Apply the Definition of the Limit

Applying the function and the limit into the expression gives us $|b/n - 0| < \epsilon$. This simplifies to $|b/n| < \epsilon$ which equals $|b|/n < \epsilon$. Now we need to find a number $N$ such that this inequality would hold true for any $\epsilon > 0$.

Find $N$

Let's isolate $n$ to find an expression for it in terms of $b$ and $\epsilon$. We have $|b|/n < \epsilon$ which leads to $n > |b|/\epsilon$. Now since $n$ is always positive, the condition of $|b|/n < \epsilon$ is true for all $n > N = |b|/\epsilon$. This means for any $\epsilon > 0$, we can find a suitable $N$ that satisfies the condition.

Conclusion

Since for every $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $|b/n - 0| < \epsilon$, therefore, by the definition of the limit, we can conclude that $\lim_{n \to \infty} (b / n) = 0$ for any real number $b$.

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Key Concepts

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

Epsilon-Delta Definition

The epsilon-delta definition of a limit is a foundational concept in calculus. It provides a rigorous way to show that a function approaches a specific limit. Here's how it works in simple terms:

Epsilon ($\epsilon$): This represents any small positive number. You can think of it as the degree of closeness you want the function to be to the limit.
Delta ($\delta$): In more general limit problems (not this specific sequence problem), delta would be the small interval around the point at which you're taking the limit.

For the specific exercise of proving $\lim (b/n)=0$, we only use epsilon because we're dealing with sequences. A sequence approaches a limit if, beyond some point in the sequence, all terms are within epsilon of the limit.
This means we can make $|b/n - 0| < \epsilon$ true for all terms by choosing a sufficiently large "N." This shows that no matter how small you choose $\epsilon$, there's some point in the sequence afterwards that stays within that range.

Real Numbers

Real numbers ($\mathbb{R}$) are the numbers we use every day. They include rational numbers, like fractions, and irrational numbers, like the square root of 2 or pi.

Properties: Real numbers are used to measure continuous quantities. They can be represented on a number line that stretches indefinitely in both positive and negative directions.
Relevance in Limits: In this exercise, any real number $b$ can be shared over increasing values of $n$. Dividing by larger numbers makes the value smaller, approaching 0.

Understanding real numbers helps in grasping how values infinitely close to a particular number can exist and be proven mathematically. This makes it easier to comprehend limits and convergence in mathematical sequences.

Convergence of Sequences

Convergence is a key concept in calculus and analysis, referring to how a sequence approaches a specific value as it progresses.

What is Convergence? A sequence converges to a limit if, as you follow the sequence further, the terms become closer to a defined number. For example, $b/n$ gets smaller as $n$ increases, converging to 0.
Importance: Understanding convergence helps in studying the behavior of sequences, ensuring that even at infinity, sequences have a predictable outcome.

In the exercise, the sequence $b/n$ was shown to converge to 0 for any real $b$. The concept of convergence assures that beyond a certain point in the sequence, all terms remain close to 0, illustrating how limits are practically achieved.

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91影视

For any \(b \in \mathbb{R}\), prove that \(\lim (b / n)=0\).

Short Answer

Step by step solution

Setup the Problem

Apply the Definition of the Limit

Find \(N\)

Conclusion

Key Concepts

Epsilon-Delta Definition

Real Numbers

Convergence of Sequences

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