Find an increasing sequence of closed sets in Rn whose union

Find an increasing sequence of closed sets in R^n whose union is open. Again, n must be at least 2.

Solution

Let A_n = [1/n,11/n] . Then A₁= [1,0], A₂= [1/2, ½], A₃= [1/3, 2/3} and so on. Also,

_n=1 A_n = (0,1). Thus, the union of the increasing union of sets A₁,A₂,…,A_n, … is an open set.