Answer to Question #5738 in Real Analysis for Akhtar Rasool

Question

Find an example in which the infinite union of closed sets
1) is not closed
2) is closed.

Answer 1

Find an example in which the infinite union of closed sets

1) is not closed

For each positive integer n, let Cn = [1 - 1/n, 2]. Clearly, each En is a closed set.
It can be checked that union (n=1 to infinity) Cn = (1, 2], which is not closed.

2) is closed

Let Cn = [-n, n] for all positive integers n. Each set is closed and the union is the set of all real numbers which is closed.