suppose Axiom C holds.
we prove proposition 1.
suppose L and U are non empty subsets of R such that L
U = R , x is in L and y is in U
==> x < y.
in other words every member of L is less than every member of U.
i.e. every member of U is an upper bound of L.
so, L is a non empty subset of real numbers which is bounded above.
so, by axiom C , L has the greatest element . let it be α. then x < α ==> x is in L and any element L .
if y > α , then y is in U.