Learning targets (75 minutes).

  1. Explain why \(\mathbb{Q}\) is incomplete: Cauchy does not imply convergent in \(\mathbb{Q}\).
  2. State the construction \(\mathbb{R}:=C_{\mathbb{Q}}/\sim\) with \( (a_n)\sim(b_n)\iff a_n-b_n\to 0\).
  3. Define addition/multiplication on classes and explain “well-defined”.
  4. Describe (at a theorem level) order and completeness (LUB).

Board plan (keep it lean).

  • (B1) Cauchy definition in \(\mathbb{Q}\).
  • (B2) \( (a_n)\sim(b_n)\iff a_n-b_n\to 0\).
  • (B3) \(\mathbb{R}=C_{\mathbb{Q}}/\sim\) and \(q\mapsto[(q,q,\dots)]\).
  • (B4) \(x+y=[(a_n+b_n)]\), \(xy=[(a_nb_n)]\); “well-defined”.
  • (B5) Completeness: every Cauchy converges / LUB property.

Active learning rhythm.

  • After each section: 30–60s “Stop & check”.
  • Do exactly one complete technical proof in class: well-definedness of addition.
  • Everything else: key identity + where boundedness enters.

Instructor pacing suggestion. 0–8 motivation; 8–23 Cauchy; 23–35 equivalence; 35–48 definition; 48–63 well-definedness; 63–72 order; 72–75 exit ticket.

Attribution. Instructional adaptation of Todd Kemp, Cauchy’s Construction of \( \mathbb{R} \) (Feb 1, 2016), UC San Diego course notes.

Goal: use collapsible frames + checks for understanding to support in-class learning.

Deep idea. \( \mathbb{R} \) is obtained by completing \( \mathbb{Q} \): we add exactly what is needed so every rational Cauchy sequence converges—by turning Cauchy behavior into numbers.

We access real numbers through sequences of rational approximations. Different approximation processes can target the same “number,” so we must identify sequences by their eventual (tail) behavior.

Example. Two approximation families for \(\pi\):

  • Decimal truncations: \( 3, 3.1, 3.14, 3.141, 3.1415, \ldots \)
  • Continued fraction convergents: \( \tfrac{22}{7}, \tfrac{355}{113}, \ldots \)

Crucial design choice. Early terms can be arbitrary; the construction only depends on what happens eventually.

Definition (Cauchy). A rational sequence \( (a_n) \) is Cauchy if for every \( \omega>0 \) there exists \(N\) such that for all \(m,n>N\), \( |a_n-a_m|<\omega \).

Theorem. If \( (a_n) \) converges in \(\mathbb{Q}\), then it is Cauchy.

Proof idea: choose \(N\) so that \( |a_n-q|<\omega/2\) for \(n>N\), then triangle inequality gives \( |a_n-a_m|<\omega\).

Theorem. Every Cauchy sequence in \(\mathbb{Q}\) is bounded.

Use the tail bound \( |a_n-a_{N+1}|<1\) and absorb finitely many terms into a maximum.

Checkpoint. In \(\mathbb{Q}\), there exist Cauchy sequences that do not converge in \(\mathbb{Q}\) (this is incompleteness).

Stop & check (60s). Give a bounded sequence that is not Cauchy. (Example: \( (-1)^n \).)

We will identify sequences that are “eventually indistinguishable.” This requires an equivalence relation.

Definition. A relation \(\sim\) on \(S\) is an equivalence relation if it is reflexive, symmetric, and transitive.

Equivalence classes. For \(s\in S\), \([s]=\{t\in S: t\sim s\}\). The classes partition \(S\).

Mini-task. On \(\mathbb{Z}\), \(a\sim b\iff a-b\) is even. How many equivalence classes are there?

Definition. Let \( C_{\mathbb{Q}} \) be the set of all rational Cauchy sequences.

Definition (Cauchy equivalence).

\[ (a_n)\sim(b_n) \iff a_n-b_n\to 0. \]

Definition (The reals).

\[ \mathbb{R}:=C_{\mathbb{Q}}/\sim. \]

A real number is an equivalence class \(x=[(a_n)]\).

Embedding. Identify \(q\in\mathbb{Q}\) with the constant class \( [(q,q,q,\ldots)] \in \mathbb{R}\).

Checkpoint. We have a set. Next we must define arithmetic, order, and then prove completeness.

Definition. If \(x=[(a_n)]\) and \(y=[(b_n)]\), define

\[ x+y := [(a_n+b_n)], \qquad xy := [(a_nb_n)]. \]

Well-definedness is the central technical obligation. A real number is a class, not a specific sequence. Operations must not depend on the chosen representative.

Do this one fully in class.

Multiplication. Key identity + boundedness of Cauchy sequences.

Stop & check (45s). Why do we need boundedness for multiplication but not for addition?

Definition (Positive). \(x\) is positive if \(x\neq 0\) and for some representative \(x=[(a_n)]\), there exists \(N\) such that \(a_n>0\) for all \(n>N\).

Order. Define \(x>y\) iff \(x-y\) is positive.

Instructor note. A standard lemma: for \(x\neq 0\), either \(x\) is eventually positive or eventually negative (after choosing a suitable representative). This supports trichotomy.

Structural Approximation Bridges \(\mathbb{Q}\) ↔ \(\mathbb{R}\)

  • Archimedean: if \(s,t>0\), then \(\exists m\in\mathbb{N}\) with \(ms>t\).
  • Density: between any two reals there is a rational.

Stop & check (45s). Use density to explain why “dense” does not mean “equal.”

The target theorem. Every nonempty \(S\subseteq\mathbb{R}\) bounded above has \(\sup S\in\mathbb{R}\).

Bisection construction (two sequences). Start with an upper bound \(u_0\) and a witness \(\ell_0\in S\). Let \(m_n=(u_n+\ell_n)/2\). If \(m_n\) is an upper bound, set \(u_{n+1}=m_n\); otherwise set \(\ell_{n+1}=m_n\). Then \(u_n-\ell_n\to 0\).

Instructor hint. Treat this section as “the punchline”: completeness is earned, not assumed.

  • \(\mathbb{R}\) is \(C_{\mathbb{Q}}/\sim\), where \( (a_n)\sim(b_n)\iff a_n-b_n\to 0\).
  • Arithmetic is termwise, but must be well-defined on classes.
  • Order can be defined via eventual positivity (requires well-definedness).
  • Completeness (LUB / every Cauchy converges) becomes a theorem.

Conceptual punchline. Limits are structural: we define \(\mathbb{R}\) precisely so that Cauchy behavior has a home where it converges.

Answer quickly (no notes).

  1. State the equivalence relation used in the construction.
  2. In one sentence: what does “well-defined” mean for addition?
  3. Give one formulation of completeness for \(\mathbb{R}\).

Fast collection idea. Ask for a 10-second verbal answer for each item, then move on.

Credits. Adapted (pedagogically) from Todd Kemp, Cauchy’s Construction of \( \mathbb{R} \) (Feb 1, 2016), UC San Diego course notes.