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0.999...
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U-limit of a sequence
« on: Apr 29th, 2012, 10:02am » |
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I am currently reading Set Theory by Thomas Jech (3rd Millennium ed.), and did not like my solution to exercise 7.6:
Let U be an ultrafilter on  and let <an> be a bounded sequence of countably infinitely many real numbers. Prove that there exists a unique U-limit a such that for every  >0, the set {n: |an-a| < }  U.
Here's my proof. If anyone could either lead me toward something more elegant (e.g. a way to link both cases, perhaps a different line of reasoning in Case I would do that) or confirm that this is the best I can do, I would be grateful.
| hidden: | Case I: U contains only infinite sets (i.e. U is nonprincipal). Since the given sequence is bounded, it has one or more cluster points. Define a mapping from pairs (c, ) to subsets of : A(c, ) = {n: |an-c| < }, where c is a cluster point and > 0. For the sake of contradiction, suppose there is > 0 such that A(c, )  U for all cluster points c. Since A(c,  )  A(c, ) when < , there exists r such that the property holds for all < r and for no > r. Furthermore, A(c,r) U; otherwise, U would be a principal ultrafilter. Choose > r such that < 3r. The set {n: r <= |an-c| < } is in U, and so is infinite, which implies that there is a cluster point c' and 0< <= (-r)/2 < r such that A(c', ) U. This is a contradiction.
As > 0 gets smaller, the cluster points c such that A(c, ) U are required to be closer to to each other (|c-c'|/2 <= ). Thus, there is a unique cluster point which satisfies the U-limit criterion. Of course, no other point x does, since for some the set {n: |an-x|< } is finite.
Case II: U contains a singleton set {m}. It is clear that {n: |an-an| < }  {m}, and thus is in U. |
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| « Last Edit: May 1st, 2012, 6:17am by 0.999... » |
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