This note concerns the equivalence between the three properties usually taken as an axiom in synthetic constructions of the real numbers. We start with the least upper bound property, call **L**, which is usually appeared in construction of the real numbers.

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L, least upper bound axiom): If is a non-empty subset of , and if has an upper bound, then has a least upper bound , such that for every upper bound of , there holds .

The second property, call **C**, is the completeness of reals.

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C, completeness axiom): If and are non-empty subsets of with the property for any and , then there is some such that for any and .

The third, also last, property, call **AC**, is the set of two results: the Archimedean property and Cantor’s intersection theorem. These two results often appear as consequences of the construction of reals.

Archimedean property: For any real numbers and with , there exists some natural number such that .

Cantor’s intersection theorem: A decreasing nested sequence of non-empty, closed intervals in has a non-empty intersection.

Our aim is to prove that in fact the above three properties (L), (C), and (AC) are equivalent. Our strategy is to show the following direction:

(L) ⟶ (C) ⟶ (AC) ⟶ (L).

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