Equivalency of some advanced calculus properties

[hsp]

i really don't know how to prove that the following are equivalent:

 

[HallsofIvy]

Did you leave something out? Prove what are equivalent?

 

[Gib Z]

Lol this looks like a scary ghost thread, with a missing passage from the OP and a double identical post from a moderator :D

 

[hsp]

equivalence of properties

Gud afternun. I have this problem to prove that these following properties are equivalent:
Nested Interval Property
Bolzano-Wierstrass theorem
Monotonic sequence property
LUB property
Heine-Borel theorem
archimedean property and cauchy convergence
line connectedness
dedekind completeness

thank you in advance.

 

[Gib Z]

Are you quite sure that they are equivalent..

 

[hsp]

yes. I've seen the diagram/sketch of the proof on the book advanced calculus by Buck.

 

[HallsofIvy]

I will confess to being not absolutely certain what "Archimedean convergence" is!

You might find this useful. I wrote it several years ago and would probably it somewhat differently now.

http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/Reals.pdf

 

[mikecon0523]

Answer for HSP

I have an answer for that.. LET US PRAY!

 

[ZioX]

I will confess to being not absolutely certain what "Archimedean convergence"
http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/Reals.pdf

You interpreted it wrong, it seems. It is archimedean property and not archimedean convergence. Archimedean property is the fact that the reals contain no 'infinitesimals'. Disregarding the formal use of infinitesimal we can just say 'for any real number x there exists a natural number n such that n>x.

'The non-existence of nonzero infinitesimal real numbers is intuitively obvious. In axiomatic theory of real numbers, it is implied by the least upper bound property as follows. Denote by Z the set consiting of all positive infinitesimals, together with zero. This set is non-empty and is bounded above by 1 (or by any other positive non-infinitesimal, for that matter) and nonempty. Therefore, Z has a least upper bound c. Suppose that the real number c is positive. Is c itself an infinitesimal? If so, then 2c is also an infinitesimal (since n(2c) = (2n)c < 1), but that contradicts the fact that c is an upper bound of Z (since 2c > c when c is positive). Thus c is not infinitesimal, so neither is c/2 (by the same argument as for 2c, done the other way), but that contradicts the fact that among all upper bounds of Z, c is the least (since c/2 < c; but every x > c/2 can't be infinitesmal: nx > nc/2 > 1). Therefore, c is not positive, so c = 0 is the only infinitesimal.'

 

[hsp]

I just want to clear my statement. What I've said is "archimedean property and cauchy convergence property"

 

[HallsofIvy]

A number of years ago I posted this:
http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/$file/Reals.pdf

which has proofs of several of those.

 

[itspixiejem]

mikecon and hsp, I think i now both of you...=)) I also need answers for this topic. Please share! =))