To gain a real understanding on real numbers.
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.
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, nonempty subset of $\Q$.
- $C$ has no maximum.
- $C$ “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.