Consider the collection of sets (0, 1/j) for all j > 0. What is the intersection of all of these sets ?The intersection of all intervals (0, 1/j) is empty. To see this, take any real number x. If x 0 it is not in any of the intervals (0, 1/j), and hence not in their intersection. If x > 0, then there exists an integer N such that 0 < 1 / N < x. But then x is not in the set (0, 1 / N) and therefore x is not in the intersection. Therefore, the intersection is empty.
Note that this is an intersection of 'nested' sets, that is sets that are decreasing: every 'next' set is a subset of its predecessor.