The strange difference between “divergent sequences” in real analysis and abstract algebra
I have been working on Commutative Algebra. A lot of the initial proofs that I’ve come across use Zorn’s lemma. The statement of Zorn’s lemma is simple enough (which I have blogged about before):
Suppose a partially ordered set has the property that every chain (i.e. totally ordered subset) has an upper bound in . Then the set contains at least one maximal element.
I came across the proof of the fact that every field has an algebraic closure, which also uses Zorn’s lemma. The argument was: continue adding field extensions infinitely. So the sequence that we have generated is something like (ad infinitum). Now is also a field (or rather can be made into a field). Also, is the limit of the sequence. We then argue that as is the maximal element in the chain of nested fields, it is the algebraic closure (use Zorn’s lemma here).
An analogy in real analysis would be: take a divergent sequence. Can we say anything about its limit except for the fact that it is (or )? There’s really not that much to say. Say we have two divergent sequences and for . All we can say is approaches faster than . Nothing else.
But here, we’ve taken something like a divergent sequence, and said something intelligent about it. This has to do with the fact that the limit of the divergent sequence is still a field, and satisfies all the axioms of a field, while the limit of a divergent sequence in real algebra does not act like a real number by any stretch of imagination. This is a rather strange fact, and should be noted for a full appreciation of the argument.
Also, we did not know right away that the limit of the sequence of field extensions would be a field. We had to make a minor argument that the limit is exactly the union of all fields in the sequence.