There are many equivalent versions of completeness in the real number system:
i) LUB/supremum property
ii) Monotone Convergence property
iii) Nested Interval property
iv) Bolzano Weierstrass property
v) Cauchy Criterion property
I've been able to prove: (i)$\implies$(ii)$\implies$(iii)$\implies$(iv)$\implies$(v)
I need help with
a) (v)$\implies$(i)
b) (iii)$\implies$(i)
P.S. In proving (v)$\implies$(i), we use the construction of 2 sequences by using mid-points. I am having problem with showing that they are Cauchy sequences