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Homework Statement
I am trying without success to provide a rigorous proof for the following exercise:
Show that the sum of a rational number and an irrational number is irrational.
Homework Equations
I am working from the following books:
Ethan D. Bloch: The Real Numbers and Real Analysis
and
Derek Goldrei: Classic Set Theory
Both use a Dedekind Cut approach to the construction of the real numbers (but Goldrei also uses Cauchy Sequences ... )
I am taking the definition of an irrational number as equivalent to an irrational cut as defined by Bloch as follows:
Bloch's definition of a Dedekind Cut plus a Lemma indicating the that there are at least as many of them as there are rational numbers are relevant ... and read as follows:
The Attempt at a Solution
I have been unable to make a meaningful start on this problem ...Peter*** EDIT ***
Reflecting on this problem ... it has become apparent to me that Bloch's definition of the addition of real numbers (in terms of Dedekind Cuts is relevant ... so I am providing the relevant definition ... as follows:
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