|May31-07, 06:37 PM||#1|
Analysis - Cauchy caracterisation of completeness
1. The problem statement, all variables and given/known data
In my book (Classical Analysis by Marsdsen & Hoffman), they use the monotone bounded sequence property as the completeness axiom. That is to say, they call complete an ordered field in which every bounded monotone sequence converges and they argue that there is a unique (up to order preserving field isomorphism) complete ordered field that we call the reals.
Then they clearly show that the completeness axiom is logically equivalent to the least upper bound property (if a subset of the reals is bounded above, then the supremum exists [i.e. is real]). They then start talking about Cauchy sequences and "hint" that the statement "Every Cauchy sequence converges" is also logically equivalent to the completeness axiom. That's what I want to verify.
3. The attempt at a solution
The "==>" part is already taken care of in the text because we used the completeness axiom to prove a lemma to the thm that every Cauchy sequence converges.
But I'm struggling a bit with the "<==" side in showing that every bounded monotone sequence is Cauchy.
I'll keep thinking about it an update this thread if I find something. Meanwhile, a hint would be post welcome
|May31-07, 06:50 PM||#2|
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