# Apostol Calculus Vol.1 Exercise 2 Help Requested

• danne89
In summary, the conversation is about a difficult exercise in Apostol's Calculus vol. 1 and a request for help with problem number 2. The problem states that when x is an arbitrary real number, there exist integers m and n such that m < x < n. There is some confusion about the problem and its solution, with one person suggesting that the answer should be based on the axioms of the book. However, another person points out that the problem does not mention positive integers and questions whether the set of integers is bounded below or above.
danne89
Hello! Anyone read Apostol's Calculus vol. 1. On p. 28 the exercises feels very hard. Can somebody help me with nr. 2?

can u post the prob. here?

I thought it would be useless because the answer must build on his axioms...
But here it comes:
If x is an arbitrary real number, prove that there exist postive integers such as m<x<n.

do you mean m < abs(x) < n?

And what if x = 0? danne89, what is the problem, word-for-word?

danne89 said:
I thought it would be useless because the answer must build on his axioms...
But here it comes:
If x is an arbitrary real number, prove that there exist postive integers such as m<x<n.

The problem states that if x is an arbitary real number, then there exist integers m and n such that m < x < n. The problem makes no reference to positive or otherwise. No wonder you're having such a hard time with the problem.

The negation says that there exists a real number x such that, for all integers m and n, (m > x) or (x > n). IOW, that the set of integers is bounded below or bounded above or both. Is that true?

## 1. What is the purpose of "Apostol Calculus Vol.1 Exercise 2"?

The purpose of "Apostol Calculus Vol.1 Exercise 2" is to provide practice problems and exercises for students learning calculus. It is a part of a larger textbook series by Tom M. Apostol that covers various topics in calculus.

## 2. Is "Apostol Calculus Vol.1 Exercise 2" suitable for beginners?

No, "Apostol Calculus Vol.1 Exercise 2" is not suitable for beginners. It is designed for students who already have a basic understanding of calculus and are looking for more challenging problems to solve.

## 3. Are there solutions provided for the exercises in "Apostol Calculus Vol.1 Exercise 2"?

Yes, there are solutions provided for the exercises in "Apostol Calculus Vol.1 Exercise 2". However, they are typically only available in the instructor's edition of the textbook.

## 4. How can "Apostol Calculus Vol.1 Exercise 2" help improve my understanding of calculus?

"Apostol Calculus Vol.1 Exercise 2" can help improve your understanding of calculus by providing a variety of practice problems and exercises. These problems can help you apply the concepts you have learned and deepen your understanding of the subject.

## 5. Can "Apostol Calculus Vol.1 Exercise 2" be used for self-study?

Yes, "Apostol Calculus Vol.1 Exercise 2" can be used for self-study. However, it is recommended to have a strong foundation in calculus before attempting the exercises in this book. It may also be helpful to have access to the solutions for reference and to check your work.

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