Apostol Calculus Vol.1 Exercise 2 Help Requested

[danne89]

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

 

[HackaB]

can u post the prob. here?

 

[danne89]

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.

 

[HackaB]

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

 

[honestrosewater]

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

 

[Lonewolf]

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.

 

[honestrosewater]

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?