Where does the less than or equal come from in the proof of density of Q in R?

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SUMMARY

The discussion centers on the proof of the density of rational numbers (Q) in real numbers (R) as presented in "Principles of Mathematical Analysis" by Walter Rudin. A participant seeks clarification on the derivation of the expression "m-1 ≤ nx" within the proof. The conversation highlights the importance of the Archimedean Property in establishing the existence of integers m1 and m2, leading to the conclusion that there exists an integer m such that m-1 is less than or equal to nx. Recommendations for supplementary texts include Spivak's "Calculus" and "Advanced Calculus: A Friendly Approach" for those new to analysis.

PREREQUISITES
  • Understanding of the Archimedean Property in real analysis
  • Familiarity with basic concepts of mathematical proofs
  • Knowledge of rational and real number properties
  • Basic calculus concepts (Calculus 1-3)
NEXT STEPS
  • Study the Archimedean Property in detail
  • Learn about integer properties and their applications in proofs
  • Read Spivak's "Calculus" for a foundational understanding of analysis
  • Explore "Advanced Calculus: A Friendly Approach" for practical examples and exercises
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Students of mathematical analysis, self-learners seeking to understand proofs, and individuals transitioning from calculus to real analysis.

completenoob
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Hello,

I have decided to study analysis on my own and am starting with principles of mathematical analysis by rudin.
I am having trouble understanding pg. 9 on the density of Q in R, part b.

It states:
If x \in R, y \in R and x<y the there exists a p \in Q such that x < p < y
Proof:
Since x<y, we have y-x>0 and the Archemedian Property furnishes a positive integer n such that:
n(y-x)>1
Applying the AP again, to obtain positive integers m1 & m2 such that m1>nx, m2>-nx
Then: -m2<nx<m1
Hence there is an integer m such that m-1 \le nx<m...

Can someone explain to me the last line? Where does this less then or equal come from?
 
Physics news on Phys.org
:cry:
 
MMmm...For being my first analysis book Principles of Mathematical Analysis by Rudin might be a little too hard. Should I study Apostol's Principles of Mathematical Analysis first? Then go back to Rudin's text?
 
Probably depends how much you already know.
 
Calc 1-3. No analysis experience.
 
Yah i need a text for self study. I have no experience with proofs.
 

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