arrow_back 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.