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Use the following steps to prove that every non-empty open subset of R is a union of at most countably many disjoint open intervals.
Suppose that G is a non-empty open subset of R.

1. For each a <- G let Ia be the union of all those open intervals I which contain a and are contained in G. Prove that Ia is a non-empty open interval.

2. Show that for every a, b <- G, either Ia = Ib, or Ia and Ib are disjoint.

3. Let F be the family of all those open intervals in R which equal Ia for some a <- G. By the last part distinct intervals in F are disjoint. Prove that the union of all intervals in F equals G.

4. By using the fact that the set of all rational numbers is countable, and subsets of countable sets are at most countable, prove that every collection of non-empty disjoint open intervals in R is a collection of at most countably many intervals.

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