# Real Number Construction From Dedekind Cuts

## A geometrically intuitive approach

### Goal

To gain a real understanding on real numbers.

### Analytical construction

I “swallowed” the Compleness Axiom, then I worked on exercises on $\sup$ and $\inf$, and then the $\epsilon$-$\delta$ criterion for limits, before completing $\Q$ with Cauchy sequences.

I’ve also heard about the completion of a metric space in a more general setting. My professor once said that it suffices to view this proof once throughout lifetime: the proof itself wasn’t very useful.

The basic arithmetic properties of $\R$, as an equivalence class of Cauchy sequences sharing the same limits, didn’t arouse our interests. That’s just an extension of its rational counterpart due to some arithmetic properties of limits.

### Algebraic construction

In my opinion, the construction of $\R$ from Dedekind cut is much more elegant.

$C \subseteq \Q$ is a Dedekind cut if

1. $C$ is a proper, nonempty subset of $\Q$.
2. $C$ has no maximum.
3. $C$ “goes left”: $\forall a \in C, b \in \Q$ with $b < a$, $b \in C$.

A real number is defined to be a Dedekind cut.

This is a proper definition since it doesn’t mention $\R$ at all. This is algebraically unnatural at the first time, but it captures the idea of cutting a point on a line into two halves. Some authors prefer to include the complement of a Dedekind cut by writing $(C,D)$ so that $C \sqcup D = \Q$, but I’ll omit that to save ink.

Since the basic arithmetic operations are far from my math studies in probabilies and statistics, I’ll just stop here and move to product $\sigma$-algebras the next time.

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