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...
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...