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$A$ and $B$ are well-ordered sets. $A, B \subseteq \mathbb{R}$

$ C := \{ n+m : n \in A , m \in B \} $

How do i now prove that $C$ is well ordered? It seem logical to me, but I have to prove that ever $S \subset C$ has a minimum.

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    Well, that is quite a substantial information.2012-11-29

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Hint: Let's assume that $C$ is not well-ordered, then there is in an infinite strictly decreasing sequence $c_1 > c_2 > \ldots$. Each of those is of the form $a_i + b_j$, so at least one of the $\{a_i\}_i$ or $\{b_j\}_j$ has to contain infinite strictly decreasing sequence, contradiction.

Edit: According to Andres Caicedo (as pointed out in the comments), the follow-up is a standard technique in Ramsey theory.

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    Yes, this is the usual argument. It is Ramsey theory, but analysis books tend not to mention the word. (You may want to add a remark or so to your answer, to clarify this detail.)2012-11-30