Give example AnnN family set every element of An is an open

Give example: (An)n?N family set- every element of An is an open, nonempty, bounded subset of R and for all n?N An+1 is subset of An but ?An??

Solution

We consider the family of sets defined as:- An = {x: -(1/2)^n < x < (1/2)^n }, n bleongs to N.

So every set , for n = 1,2,3,... is open as defined above.

Every set is is bounded both above and below. If x is any element belonging to a set, An , the for x in any set An , -1/2 < x < (1/2) .

Cearly the set An+1 is a subset of An as all the elements in An+1 is in An.

The intersection of all the sets cannot be a null set or empty set , as every set An has the element zero.