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The sum of Dedekind cuts alpha and beta is defined to be {a+b : a in alpha and b in beta}. But why does this correspond to the intuitive notion of sum? Why is sum defined this way? Spivak's book says it's an "obvious" definition, but it doesn't seem obvious at all to me.

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You need to intuitively understand that if $\alpha$ and $\beta$ are real numbers, and $q$ is a rational number such that $q<\alpha+\beta$ then there are rational numbers $a,b$ such that $a<\alpha$ and $b<\beta$ such that $a+b=q$.

The way to show this is to set $\varepsilon=\alpha+\beta-q$ and pick a rational number $a\in(\alpha-\varepsilon,\alpha)$. Then let $b=q-a$, and show that $b<\beta$.

So this follows from the intuition that any open interval contains a rational number.

(The other part - if $a<\alpha$ and $b<\beta$ then $a+b<\alpha+\beta$ - is more obvious.)

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Which set of the cut are you using, the upper or the lower one?

If using the upper one: every $a\in A$ is an upper bound for $x=\inf A$ and every $b\in B$ is an upper bound for $y=\inf B$. Thus $a+b$ is an upper bound for $x+y$. One still has to convince oneself that these bounds "reach all the way down" so that $x+y=\inf(A+B)$ holds.

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