Hello, I'm hoping you guys can help me unravel this apparent contradiction and show me where my reasoning went wrong. I'm working out of the book "Mathematical Analysis" by Apostol (second addition) and here a few of Apostol's definitions and notation.

B(a;r). The open ball of radius r centered at a.

Interior Point. Let S be a subset of R (reals), and assume that a is an element of S. Then a is called an interior point of S if there is an open ball with center at a, all of whose points belong to S.

Open Set. A set S in R is called open if all its points are interior points.

Closed Set. A set S in R is called close dif its complement R - S is open.

Later in the book its mentioned that the empty set is open (vacuously). In the argument I wrote, I tried to show that a countably infinite union, S, of closed sets is open. But this union should be equal to R and so the complement is the empty set which is open, implying S is closed. Here is a link to what I wrote up, you can click on the picture to magnify it so its readable:

Image of closed sets - Photobucket - Video and Image Hosting