# Ngô Quốc Anh

## October 15, 2021

### Least upper bound axiom, completeness axiom, and Archimedean property and Cantor’s intersection theorem are equivalent

Filed under: Giải Tích 1 — Ngô Quốc Anh @ 17:03

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.

(L, least upper bound axiom): If $A$ is a non-empty subset of $\mathbf R$, and if $A$ has an upper bound, then $A$ has a least upper bound $u$, such that for every upper bound $v$ of $A$, there holds $u \leq v$.

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

(C, completeness axiom): If $X$ and $Y$ are non-empty subsets of $\mathbf R$ with the property $x \leq y$ for any $x \in X$ and $y \in Y$, then there is some $c \in \mathbf R$ such that $x \leq c \leq y$ for any $x \in X$ and $y \in Y$.

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 $x$ and $y$ with $x>0$, there exists some natural number $n$ such that $nx > y$.

Cantor’s intersection theorem: A decreasing nested sequence of non-empty, closed intervals in $\mathbf R$ 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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