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 [itex]\alpha[/itex] in rational numbers , its additive inverse is given by [itex]\beta[/itex] .

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

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

No problem till this point.

Then we are supposed to prove that [itex]\alpha[/itex] + [itex]\beta[/itex] = 0.

For this ,part 1 of the proof is that any element of [itex]\alpha[/itex] + [itex]\beta[/itex] should be a negative rational.

Here's Rudin's proof

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

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