Apostol Calculus Vol.1 Exercise 2 Help Requested

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SUMMARY

The discussion revolves around Exercise 2 from Apostol's Calculus Vol. 1, which asks to prove that for any arbitrary real number x, there exist positive integers m and n such that m < x < n. Participants clarify that the problem does not specify m and n must be positive integers, and they emphasize the importance of understanding the implications of boundedness in relation to real numbers and integers. The negation of the statement is also discussed, highlighting the relationship between real numbers and the set of integers.

PREREQUISITES
  • Understanding of real numbers and their properties
  • Familiarity with basic concepts of mathematical proofs
  • Knowledge of boundedness in mathematical sets
  • Experience with Apostol's Calculus Vol. 1, particularly axioms and definitions
NEXT STEPS
  • Study the concept of bounded sets in real analysis
  • Review the axioms presented in Apostol's Calculus Vol. 1
  • Practice constructing mathematical proofs involving inequalities
  • Explore the implications of the completeness property of real numbers
USEFUL FOR

Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of real analysis and mathematical proofs.

danne89
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Hello! Anyone read Apostol's Calculus vol. 1. On p. 28 the exercises feels very hard. Can somebody help me with nr. 2?
 
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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 positive 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 positive 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?
 

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