Is There an Error in the Additive Inverse Definition in Rudin's PoMA?

[Khichdi lover]

In Rudin PoMA , chapter 1
in appendix 1 construction of real numbers as Dedekind Cut's is given.

I feel there is an error in the definition of additive inverse of a 'cut' .

given a cut \alpha in rational numbers , its additive inverse is given by \beta .

a rational p belongs to \beta , if there exists a rational number r>0 such that -p-r\notin \alpha .

the additive identity 0^* is the set of all negative rational numbers.

No problem till this point.

Then we are supposed to prove that \alpha + \beta = 0.

For this ,part 1 of the proof is that any element of \alpha + \beta should be a negative rational.Here's Rudin's proof

If r \in \alpha and s \in \beta ,
then -s \notin \alpha , hence r<-s , r+s < 0 . Thus \alpha+\beta \subset 0^*

How does -s \notin \alpha follow from s \in \beta ? I feel this is printing error.(do you agree on this?)

Anyway, the proof can be slightly changed to make it correct :-
If r \in \alpha and s \in \beta ,
then there is a rational number t>0 , such that
- s - t \notin \alpha,
hence r < -s - t ,
r + s < - t < 0.

Am I right ? I ordered the book's 3rd ed in India, and I am discovering that the book has many errors .
I checked this errata - http://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_notes.pdf , but couldn't find the error I pointed out.

*note :- if this isn't posted in right sub-forum,Please move this to appropriate forum, in which one is supposed to discuss errata*

 

[micromass]

Rudin is correct. There exist a rational t>0 such that -s-t is not in \alpha. But

-s-t&lt;-s

So -s is also not in \alpha.

 

[Khichdi lover]

Oops. Thanks.

I got confused between \notin and \in .



Doubt resolved. Please lock the thread if the need be.