Proving the sum of a rational and irrational number is irrational

[Math Amateur]

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:

The note preceding the above definition mentions Lemma 1.6.8 which reads as follows:

 

[Math Amateur]

After reflecting on this problem ... I now think that the answer to the exercise above may be disappointingly trivial ...

Consider the following:

Let $a \in$ $\mathbb{Q}$ and let $b \in \mathbb{R}$ \ $\mathbb{Q}$

Then suppose a + b = r

Now ... assume r is rational

Then b = r - a ...

But since r and a are rational ... we have r - a is rational ..

Then ... we have that an irrational number b is equal to a rational number ...

Contradiction!

So ... r is irrational ..
Is that correct?

Peter

 

[fresh_42]

Is that correct?

Yes.