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I'm not sure how to approach this proof? any ideas

Let $A$ be a set of intervals of the real line any two of which are disjoint - in other words, if $(a,b)$ and $(x,y)$ are distinct elements of $A$ then $(a,b)\cap(x,y)=\emptyset$. Prove that A is countable. *(Use the fact that $\mathbb Q$ is countable)

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    A similar question: [Every collection of disjoint non-empty open subsets of $\mathbb R$ is countable?](http://math.stackexchange.com/questions/75781/every-collection-of-disjoint-non-empty-open-subsets-of-mathbbr-is-countable).2012-06-25

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Since $\mathbb{Q}$ is dense in $\mathbb{R}$ each interval contains some positive number of rational numbers. For each interval $I_{k}$ choose such a rational number $a_k \in I_k$, and denote the set of all such $a_k$ as $N$. This set is clearly countable, as it is a subset of the rational numbers, and thus $N$ is of an equal or smaller cardinality. Note that $N$ is one to one with $A$, and thus has equal size. $A$ is therefore countable.

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    Nitpicking: "some number" should be "some positive number"2012-07-27