5.2 Cauchy sequences and another model of the reals

The previous note defined convergence by comparing sequence terms with a previously known limit $L$. This note asks a harder question:

What if we want to recognize convergence before we already know the limit as an existing real number?

That question leads to the notion of a Cauchy sequence and to a second construction of the real numbers.

Why we need an internal test for convergence

A good motivation for the new definition is a sequence of rational numbers whose terms appear to close in on some point of the line:

$$1,\qquad 1+\frac12,\qquad 1+\frac12+\frac1{3\cdot 2},\qquad 1+\frac12+\frac1{3\cdot 2}+\frac1{4\cdot 3\cdot 2\cdot 1},\ \ldots$$

Visually, the terms seem to gather around a limiting point. But if we are trying to construct the real numbers from the rationals, then we do not yet have the right to name that limiting point as an already available element of $R$.

So instead of asking whether the sequence is close to some external point $L$, we ask whether the terms are getting close to each other.

The definition of Cauchy sequence

This definition has the same quantifier shape as the limit definition, but the comparison target has changed:

• for an ordinary limit, you compare $x_n$ with a fixed number $L$;
• for a Cauchy condition, you compare late terms of the sequence with one another.

So a Cauchy sequence is one whose tail fits into narrower and narrower bands.

Why convergent sequences are automatically Cauchy

The key proposition is the following.

The proof idea is short and very important.

This theorem says that genuine convergence always forces the sequence tail to compress.

Equivalent Cauchy sequences

If Cauchy sequences are going to represent real numbers, then different sequences that “head toward the same place” should count as the same real.

This condition says that the two tails eventually lie arbitrarily close to one another. Intuitively, they are describing the same limiting point on the line.

This notion of equivalence is really an equivalence relation on the set of Cauchy sequences. The triangle inequality is the key tool for checking symmetry and transitivity.

The alternative construction of $R$

Now turn the idea into a definition.

This is a big shift in viewpoint:

• in the Dedekind-cut model, a real number is a left/right split of $Q$;
• in the Cauchy-sequence model, a real number is a whole family of rational sequences that become indistinguishable in the limit.

Neither model is “more real” than the other. They are two rigorous ways to build the same number system.

How rationals sit inside the model

It remains to say how rational numbers sit inside this construction. The answer is natural: a rational q is represented by the constant Cauchy sequence

$$(q,q,q,q,\ldots).$$

The real number is therefore not any one representative sequence by itself. It is the full equivalence class.

Why boundedness matters

Every Cauchy sequence is bounded. That fact is crucial because it allows multiplication to work cleanly.

If $(x_n)$ and $(y_n)$ are Cauchy, then:

• $(x_n+y_n)$ is Cauchy;
• $(x_ny_n)$ is Cauchy.

For multiplication, one needs a common bound $M$ so that

$$|x_my_m-x_ny_n| \le |x_m|\cdot |y_m-y_n|+|y_n|\cdot |x_m-x_n| \le M|y_m-y_n|+M|x_m-x_n|.$$

Once the two original sequences are Cauchy, the right-hand side can be made as small as desired.

Operations and order on equivalence classes

After establishing that termwise sums and products of Cauchy sequences stay Cauchy, define addition and multiplication on equivalence classes.

So if $[(x_n)]$ and $[(y_n)]$ are real numbers in this model, then:

$$[(x_n)] + [(y_n)] := [(x_n+y_n)],$$ $$[(x_n)] \cdot [(y_n)] := [(x_ny_n)].$$

Order is defined by eventual comparison of representatives: one class is below another when sufficiently late terms of a representative of the first lie below sufficiently late terms of a representative of the second.

The serious work is then to check that all these definitions are well-defined. That means the answer does not depend on which representative sequences you chose.

The conclusion of the construction

These notes do not expand every technical detail here. Some parts remain as exercises, including the well-definedness of operations and the final proof of completeness. But conceptually the message is clear:

• Cauchy sequences capture the idea of internal convergence;
• equivalence classes prevent different approximating sequences from giving different names to the same real;
• completeness is built in, because every Cauchy sequence should already have a place to land in the completed system.

Quick checks

Exercises

Related notes

Read this after 5.1 Sequences and epsilon-N limits and 4.3 Completeness and gaps in Q. Then continue to 5.3 Delta-epsilon limits, limit laws, and continuity.