Cauchy Criterion for Sequences

Introduction: Why this idea matters

Imagine students, that you are watching the numbers in a sequence get closer and closer together, even if you do not yet know what number they are heading toward 📈. In Real Analysis, this is the core idea behind the Cauchy criterion. It gives us a way to test whether a sequence is “settling down” using only the sequence itself, without first guessing its limit.

In this lesson, you will learn:

what a Cauchy sequence is,
what the Cauchy criterion says,
how to use the criterion in examples,
and why it is important for understanding completeness of $\mathbb{R}$.

This topic is a key bridge between sequences and the bigger idea that the real numbers contain all their limit points. It also helps explain why some sequences behave nicely in $\mathbb{R}$, while similar sequences may fail in other number systems.

What is a Cauchy sequence?

A sequence $(a_n)$ is called a Cauchy sequence if its terms eventually get arbitrarily close to each other. More precisely, for every $\varepsilon > 0$, there exists a natural number $N$ such that whenever $m,n \ge N$, we have

$|a_n-a_m|<\varepsilon.$

This definition is important because it talks only about the terms of the sequence, not about any possible limit.

Let’s translate the idea into everyday language. If a sequence is Cauchy, then after some point all later terms are packed tightly together. You can think of the sequence as a group of students gathering into a small circle: after enough time, everyone is standing very close to everyone else 👥.

For example, the sequence

$a_n=\frac{1}{n}$

is Cauchy in $\mathbb{R}$. Why? Because as $n$ gets larger, the numbers become smaller and closer to $0$, and any two sufficiently late terms are close to one another.

On the other hand, the sequence

$a_n=n$

is not Cauchy, because later terms become farther and farther apart. For instance, $|n-(n+1)|=1$, and more generally the gap between large terms does not become small.

The Cauchy criterion

The Cauchy criterion is the statement that helps us test convergence through closeness of terms.

For sequences of real numbers, the criterion says:

(a_n) \text{ converges in } \mathbb{R} $\iff$ (a_n) \text{ is a Cauchy sequence.}

This is a very powerful result. It says that in $\mathbb{R}$, a sequence converges exactly when its terms eventually bunch together as tightly as we want.

There are two directions here:

If $(a_n)$ converges, then it is Cauchy.
If $(a_n)$ is Cauchy, then it converges.

The first direction is true in any metric space, not just in $\mathbb{R}$. The second direction depends on completeness, which we will discuss soon.

Why convergent sequences are Cauchy

Suppose $a_n \to L$ as $n \to \infty$. Then for every $\varepsilon > 0$, there exists $N$ such that if $n \ge N$, then

$|a_n-L|<\frac{\varepsilon}{2}.$

Now if $m,n \ge N$, the triangle inequality gives

|a_n-a_m| \le |a_n-L|+|L-a_m| < \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.

So the sequence is Cauchy.

This argument is a standard proof technique in analysis: if two things are both close to the same third thing, then they are close to each other. It is a useful pattern students should remember.

Why Cauchy sequences converge in $\mathbb{R}$

The opposite direction is special. In the real numbers, every Cauchy sequence converges because $\mathbb{R}$ is complete. Completeness means that there are no “missing” limit points for Cauchy sequences.

This is not true in every number system. For example, in the rational numbers $\mathbb{Q}$, there are Cauchy sequences of rational numbers that do not converge to a rational number. A famous example is a sequence of rational approximations to $\sqrt{2}$. The sequence is Cauchy in $\mathbb{Q}$, but its limit is irrational, so it does not converge in $\mathbb{Q}$.

This difference shows why completeness is a major property of $\mathbb{R}$.

How to use the Cauchy criterion in practice

To use the Cauchy criterion, you usually need to show that for every $\varepsilon > 0$, terms far enough along in the sequence stay within $\varepsilon$ of each other.

Let’s look at some common patterns.

Example 1: A simple convergent sequence

Consider $a_n=\frac{1}{n}$.

To show it is Cauchy, take any $\varepsilon > 0$. Choose $N$ such that

$\frac{1}{N}<\varepsilon.$

Then for $m,n \ge N$, assuming $m \le n$, we have

$\left|$$\frac{1}{n}$-$\frac{1}{m}$$\right|$ \le $\frac{1}{m}$ \le $\frac{1}{N}$<\varepsilon.

So $(1/n)$ is Cauchy.

This example is useful because it shows the sequence gets closer to $0$, but the Cauchy criterion only needs the terms to get close to one another.

Example 2: A sequence that is not Cauchy

Consider $a_n=(-1)^n$.

The sequence alternates between $1$ and $-1$. No matter how far out you go, the terms do not stay close to each other. In fact, if $m$ and $n$ have opposite parity, then

$|a_n-a_m|=2.$

So the sequence cannot be Cauchy. Since every convergent sequence is Cauchy, this also shows the sequence does not converge.

Example 3: Using a known bound

Suppose we define

$a_n=\sum_{k=1}^n \frac{1}{2^k}.$

These are partial sums of a geometric series. For $m>n$,

|a_m-a_n|=$\sum_{k=n+1}$^m $\frac{1}{2^k}$ \le $\sum_{k=n+1}$^{$\infty$} $\frac{1}{2^k}$ = $\frac{1}{2^n}$.

Since $\frac{1}{2^n}\to 0$, the sequence is Cauchy.

This example shows a common strategy: estimate the difference between two later terms by a tail sum that becomes small.

Connection to completeness of $\mathbb{R}$

The Cauchy criterion is closely tied to completeness. A metric space is complete if every Cauchy sequence converges in that space.

For the real numbers, completeness is one of the most important foundational facts in analysis. It guarantees that if numbers get arbitrarily close together in the Cauchy sense, then there is an actual real number they approach.

This is why many arguments in analysis can be built from Cauchy sequences. Instead of guessing a limit directly, you prove the sequence is Cauchy, and completeness gives the limit for free.

This idea also helps explain why certain proofs work in $\mathbb{R}$ but fail in $\mathbb{Q}$. A sequence can be Cauchy in the rational numbers and still “want” to converge to a number outside $\mathbb{Q}$. The real numbers fill in those gaps.

Real-world style intuition

Think of measuring the temperature in a room every minute. If the readings from later and later minutes differ by less than any tiny amount you choose, then the data are Cauchy-like. In a complete setting like $\mathbb{R}$, we can say the measurements approach a single real number. That does not mean the room has stopped changing completely, but it does mean the measurements are stabilizing in a precise mathematical sense 🌡️.

Cauchy sequences versus bounded sequences

A common mistake is to think that every bounded sequence must be Cauchy. This is false.

For example, the sequence $(-1)^n$ is bounded because all terms are either $1$ or $-1$. But it is not Cauchy, since the terms never get close together.

So boundedness alone is not enough. A sequence may stay trapped in a small interval and still keep jumping around. The Cauchy criterion is stronger: it demands internal closeness among the later terms.

However, there is an important related theorem: every Cauchy sequence in $\mathbb{R}$ is bounded. This is often useful in proofs. If a sequence is Cauchy, then after some point all its terms lie within $1$ of one another, and the early terms are only finitely many values, so the whole sequence is bounded.

The role of the Cauchy criterion in Real Analysis

The Cauchy criterion is a central tool because it converts a convergence problem into an estimate problem. Instead of finding the limit first, you prove the sequence becomes internally consistent.

This criterion fits into a larger structure:

Cauchy sequences describe “potential limits.”
Completeness of $\mathbb{R}$ guarantees those potential limits really exist in $\mathbb{R}$.
The Bolzano-Weierstrass theorem gives another way to study bounded sequences by finding convergent subsequences.

These ideas work together throughout Real Analysis. When students studies series, function sequences, or metric spaces, Cauchy reasoning often appears as a key step.

Conclusion

The Cauchy criterion says that in $\mathbb{R}$, a sequence converges exactly when its terms eventually become arbitrarily close to one another. This is one of the most important ideas in Real Analysis because it connects the behavior of a sequence to the completeness of the real numbers.

A convergent sequence is always Cauchy, and in $\mathbb{R}$ every Cauchy sequence converges. This is not true in incomplete spaces like $\mathbb{Q}$, which shows why completeness matters.

If students remembers only one big idea from this lesson, it should be this: in a complete number system, “getting close together” eventually means “approaching a real limit.” That is the heart of the Cauchy criterion ✅.

Study Notes

A sequence $(a_n)$ is Cauchy if for every $\varepsilon > 0$, there exists $N$ such that $|a_n-a_m|<\varepsilon$ for all $m,n \ge N$.
The Cauchy criterion in $\mathbb{R}$ says that $(a_n)$ converges if and only if it is Cauchy.
Every convergent sequence is Cauchy in any metric space.
The converse, that every Cauchy sequence converges, is the definition of completeness.
$\mathbb{R}$ is complete, but $\mathbb{Q}$ is not.
A Cauchy sequence must be bounded, but a bounded sequence need not be Cauchy.
Common proof strategy: estimate $|a_n-a_m|$ for large $m,n$ and make it smaller than $\varepsilon$.
Examples of Cauchy sequences in $\mathbb{R}$ include $\left(\frac{1}{n}\right)$ and partial sums of convergent series.
Examples of sequences that are not Cauchy include $(-1)^n$ and $n$.
The Cauchy criterion is a key tool for studying convergence, completeness, and limit behavior in Real Analysis.