Proving the Density of a Specific Set in Real Numbers

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SUMMARY

The discussion centers on proving that the set \{x ∈ ℝ: x = p + qξ, p, q ∈ ℤ\} is dense in the real numbers ℝ for any irrational number ξ. A proof utilizing concepts from measure theory was initially sought, but the contributor later concluded that the proof is straightforward. This indicates that a simpler approach exists, which does not require advanced knowledge of measure theory.

PREREQUISITES
  • Understanding of real numbers and their properties
  • Familiarity with irrational numbers and their characteristics
  • Basic knowledge of algebraic expressions involving integers
  • Introductory concepts in measure theory (optional for deeper understanding)
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  • Study the concept of density in real numbers
  • Explore proofs involving linear combinations of irrational numbers
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Mathematicians, students studying real analysis, and anyone interested in the properties of irrational numbers and their applications in density proofs.

D.K.
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Is there an easy way to prove that for any irrational [tex]\xi[/tex] the set:

[tex]\{x \in \mathbb{R}: x = p + q\xi, \ p, q \in \mathbb{Z}\}[/tex] is dense in [tex]\mathbb{R}[/tex]?

I know a proof involving notions from measure theory of which I unfortunately know nothing about. Any help would be very appreciated.
 
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Nevermind, it turned out to be rather easy.
 

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