Show that (a,b) = ( ) , where , { } is a decreasing sequence of rational numbers with , { } is an increasing sequence of rational numbers with .
Suppose x is in (a, b). That is, that a< x< b. Show that, for some n, . That should be easy. Take . Since , there exist n such that so . Since , there exist m such that so .
It should be clear that is x< a, then x is in NONE of and so not in their union. Similarly for x> b. The important part is x= a and x= b. You need to show that and for all n.