Consider the subsets of R defined as follows: An is the interval (0,1/n) for all natural numbers. Show that:
U (union) n=1 to infinity of An = (0,1)
Here's what I tried as a proof:
Show that (0,1) is a subset of U n=1 to infinity An
Take an arbitrary real number a in (0,1) so a is in the set {x such that there exists some n such that x is in (0,1/n) for all natural numbers n}
Show that U n=1 to infinity is a subset of (0,1)
Take x in U (0,1/n)
By definition of union this means there exists some n such that x is in (0,1/n) for all natural numbers n
I'm not sure if this is the right way to go? Or should I prove if the set is countable?
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