Prove that the following are equivalent. They are all true for
but not for . Give counter-examples for each one in .
The Least-Upper-Bound (LUB) principle, aka Dedekind Completeness: If has an upper bound then it has a least-upper-bound (supremum) Dual of (1): If has a lower bound then it has a greatest-lower-bound (infimum) The Monotone Sequence Theorem: Any bounded (from above) monotone increasing sequence has a limit (and this limit is the supremum of this sequence)
Dual of (3): Any bounded (from below) monotone decreasing sequence has a limit (and this limit is the infimum of this sequence)
Bolzano-Weierstrass [1]: Every infinite bounded set has a point of accumulation (an element is a point of accumulation of if for any there is an infinite subset of which is -close to ) Bolzano-Weierstrass [2]: Every bounded sequence has a convergent subsequence
Metric Completeness: Any Cauchy sequence converges (check wikipedia for this definition) [Why are Cauchy sequences important?]
Cantor's Intersection Theorem: Denote by the set of all number that are and . Assume there is a monotone increasing sequence and a monotone decreasing sequence such that for any , . Then there is some number such that for all n. (Another nice way of putting this is that ) Half-Line Theorem (non-standard name): Let such that for every and , then . Then is of the form or for some
Categorization of the Axioms
(1, 2) are duals so we unite them.
(3, 4) are duals so we unite them.
We are left with the following axioms: 1, 3, 5, 6, 7, 8, 9.
We split them into two types -- set axioms and sequence axioms
Set Axioms: 1, 5, (8), 9
Sequence Axioms: 3, 6, 7, (8)
It is easy to show that
, etc... The hard part is to prove that one of the sequence axioms
one of the set axioms.
Todo:
Find a counterexample for (1) in [DONE] Understand (5), (7) and (8)
Think how you would prove (1) [or any other set axiom]
Solution
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