Equivalence of 8 properties in Real Analysis

[itspixiejem]

Please help me prove that the following properties are equivalent Nested Interval Property
Bolzano-Wierstrass theorem
Monotonic sequence property
LUB property
Heine-Borel theorem
archimedean property and cauchy sequence
line connectedness
dedekind completeness

Please help! Thanks!

 

[micromass]

What have you tried already?

 

[itspixiejem]

I have already drawn the diagram to connect these properties.

Dedekind completeness ---> LUB property --> line connectedness ---> Dedekind completeness

LUB property --> Heine-Borel Theorem --> Nested Interval property

Nested interval --> Bolzano Weierstrass --> Cauchy sequence

Bolzano-Weierstrass --> Monotonic Sequence --> LUB Property --> Monotonic Sequence

LUB Property --> Archimedian property+Cauchy sequences --> Monotonic Sequence

Just connect them all to make one diagram. I do not know how to attach picture here.

I am still trying to prove some implications... =(

 

[micromass]

But you have all the implications already!
What do you think you're missing?

 

[itspixiejem]

I need to prove all those implications. =(

 

[micromass]

But you already did!

What implication do you think you miss?

 

[HallsofIvy]

micromass, where did he show any proof at all?

itspixiejem, those are all difficult proofs. Some are given in texts- for example most Calculus texts show that LUB implies monotone convergence. And it is not to difficult to show that monotone convergence implies that every bounded sequence has a convergent subsequence (Bolzano-Weierstrasse). And from that, you can show that every Cauchy sequence converges by showing that a Cauchy sequence must be bounded.

 

[itspixiejem]

I haven't proven any of those applications yet. I've just drawn the diagram.

 

[itspixiejem]

Thanks HallsofIvy. I'll try to decipher what you've written. I think I still have to read more... By the way, I am a SHE... =))

 

[micromass]

Oh, I'm so sorry! I thought the diagram meant what you already proved.

The best thing to do now is check your real analysis book to see which proofs they already have. This will fill in some gaps. Afterwards, we'll see what remains to be done...