### 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

- $C$ is a
proper, nonemptysubset of $\Q$.- $C$ has
nomaximum.- $C$ “goes left”: $\forall a \in C, b \in \Q$ with $b < a$, $b \in C$.
A

real numberis 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.