I have no background in pure mathematics, and I'm trying to learn how to be more rigorous in general. To help with this, I am trying to make everything more explicit as I progress through Rudin's Principles of Mathematical Analysis. I am currently going through theorem 1.19's proof where Rudin constructs $\mathbb{R}$ from $\mathbb{Q}$.

In particular, in step 5, Rudin states that it is obvious that if $\alpha , \beta , \gamma \in \mathbb{R}$ and $\alpha < \beta$, then $\alpha + \gamma < \beta + \gamma$.

So far, the book has defined $\alpha , \beta , \gamma$ as subsets of $\mathbb{Q}$ called cuts, and has shown that cuts respect the field axioms of addition. I am intuitively convinced that the set $\beta + \gamma$ contains elements that $\alpha + \gamma$ does not, but am struggling with how to formalize it. The following is my thought process on my (failed) attempt.

I know that: $\beta > \alpha \implies \exists x \in \beta : x \notin \alpha$, and I think the next step is the take some value $y \in \gamma$ and show that $y + x \in \beta + \gamma$ while $y + x \notin \alpha + \gamma$. I originally thought this would work by taking $y = \sup \gamma$, but $\sup \gamma$ is a set, not a rational number. When I try to think about taking $y < \sup \gamma$ (edit: sorry, this should say $y \in \gamma$), it's clear to me that $y + x \in \beta + \gamma$, but it's no longer clear to me that $y + x \notin \alpha + \gamma$. Any help would be greatly appreciated!

I have no background in pure mathematics, and I'm trying to learn how to be more rigorous in general. To help with this, I am trying to make everything more explicit as I progress through Rudin's Principles of Mathematical Analysis. I am currently going through theorem 1.19's proof where Rudin constructs $\mathbb{R}$ from $\mathbb{Q}$.

In particular, in step 5, Rudin states that it is obvious that if $\alpha , \beta , \gamma \in \mathbb{R}$ and $\alpha < \beta$, then $\alpha + \gamma < \beta + \gamma$.

So far, the book has defined $\alpha , \beta , \gamma$ as subsets of $\mathbb{Q}$ called cuts, and has shown that cuts respect the field axioms of addition. I am intuitively convinced that the set $\beta + \gamma$ contains elements that $\alpha + \gamma$ does not, but am struggling with how to formalize it. The following is my thought process on my (failed) attempt.

I know that: $\beta > \alpha \implies \exists x \in \beta : x \notin \alpha$, and I think the next step is the take some value $y \in \gamma$ and show that $y + x \in \beta + \gamma$ while $y + x \notin \alpha + \gamma$. I originally thought this would work by taking $y = \sup \gamma$, but $\sup \gamma$ is a set, not a rational number. When I try to think about taking $y < \sup \gamma$ (edit: sorry, this should say $y \in \gamma$ not $y < \sup \gamma$), it's clear to me that $y + x \in \beta + \gamma$, but it's no longer clear to me that $y + x \notin \alpha + \gamma$. Any help would be greatly appreciated!

I have no background in pure mathematics, and I'm trying to learn how to be more rigorous in general. To help with this, I am trying to make everything more explicit as I progress through Rudin's Principles of Mathematical Analysis. I am currently going through theorem 1.19's proof where Rudin constructs $\mathbb{R}$ from $\mathbb{Q}$.

In particular, in step 5, Rudin states that it is obvious that if $\alpha , \beta , \gamma \in \mathbb{R}$ and $\alpha < \beta$, then $\alpha + \gamma < \beta + \gamma$.

So far, the book has defined $\alpha , \beta , \gamma$ as subsets of $\mathbb{Q}$ called cuts, and has shown that cuts respect the field axioms of addition. I am intuitively convinced that the set $\beta + \gamma$ contains elements that $\alpha + \gamma$ does not, but am struggling with how to formalize it. The following is my thought process on my (failed) attempt.

I know that: $\beta > \alpha \implies \exists x \in \beta : x \notin \alpha$, and I think the next step is the take some value $y \in \gamma$ and show that $y + x \in \beta + \gamma$ while $y + x \notin \alpha + \gamma$. I originally thought this would work by taking $y = \sup \gamma$, but $\sup \gamma$ is a set, not a rational number. When I try to think about taking $y < \sup \gamma$, it's clear to me that $y + x \in \beta + \gamma$, but it's no longer clear to me that $y + x \notin \alpha + \gamma$. Any help would be greatly appreciated!

I have no background in pure mathematics, and I'm trying to learn how to be more rigorous in general. To help with this, I am trying to make everything more explicit as I progress through Rudin's Principles of Mathematical Analysis. I am currently going through theorem 1.19's proof where Rudin constructs $\mathbb{R}$ from $\mathbb{Q}$.

In particular, in step 5, Rudin states that it is obvious that if $\alpha , \beta , \gamma \in \mathbb{R}$ and $\alpha < \beta$, then $\alpha + \gamma < \beta + \gamma$.

So far, the book has defined $\alpha , \beta , \gamma$ as subsets of $\mathbb{Q}$ called cuts, and has shown that cuts respect the field axioms of addition. I am intuitively convinced that the set $\beta + \gamma$ contains elements that $\alpha + \gamma$ does not, but am struggling with how to formalize it. The following is my thought process on my (failed) attempt.

I know that: $\beta > \alpha \implies \exists x \in \beta : x \notin \alpha$, and I think the next step is the take some value $y \in \gamma$ and show that $y + x \in \beta + \gamma$ while $y + x \notin \alpha + \gamma$. I originally thought this would work by taking $y = \sup \gamma$, but $\sup \gamma$ is a set, not a rational number. When I try to think about taking $y < \sup \gamma$ (edit: sorry, this should say $y \in \gamma$), it's clear to me that $y + x \in \beta + \gamma$, but it's no longer clear to me that $y + x \notin \alpha + \gamma$. Any help would be greatly appreciated!