Infinite Unions. It isn't enough to say "You just take your infinitely many sets and union them together". There's a problem with setting a task (putting these sets together) that has infinitely many steps. A person, or a machine, can be asked to do any finite number of tasks (even though, in practice, it might take a long time for them all to be done!), but how can we imagine a person completing infinitely many steps!

That's why we need a clear definition of the union of infinitely many things, which can be phrased in such a way that the set is described for us in a single line, rather than as an instruction to go away and "do" infinitely many things.

Also notice that the union of infinitely many sets is accomplished in a single step. We write down a description of a set in terms of a single criterion which determines whether or not any given element belongs to the set. We don't take any sort of a limit. There is no "approximating sequence" of sets that gets closer and closer to the final answer. The answer is just immediately there.

Analysis WebNotes by John Lindsay Orr.
Comments to the author: jorr@math.unl.edu

All contents copyright (C) 1996 John L. Orr
University of Nebraska--Lincoln
All rights reserved

Last modified: May 1996