Prove that a nonempty, bounded closed set $S$ in $R^1$ can be obtained from a closed interval by removing a countable disjoint collection of open intervals whose endpoints belong to $S$
Here is the exercise, where the Representation Theorem for Open Sets on The Real Line is used.
Here i have two questions:
1)Why $R^1 - S = [\inf S,\sup S] - S$ or $R^1 = [\inf S,\sup S]$?
2)I understand proof of the Representation Theorem for Open Sets but it's a bit hard to visualize it. Can you provided some picture of this theorem, if it is possible?