Scientific claim

Cauchy Sequence Definition

Claim

Let $(X,d)$ be a metric space, let $\mathbb{N}=\{1,2,3,\ldots\}$ be the set of natural numbers, and let $(x_n)_{n\in\mathbb{N}}$ be a sequence of points in $X$ .

The sequence $(x_n)_{n\in\mathbb{N}}$ is a Cauchy sequence if, for every positive real number $\varepsilon>0$ , there exists a natural number $N\in\mathbb{N}$ such that, whenever $n,m\in\mathbb{N}$ satisfy $n\geq N$ and $m\geq N$ :

d(x_n,x_m)<\varepsilon.

Theoretical: Not applicableExperimental: Not applicable

Assumptions

No assumptions recorded.

Proof

This is a definition, so it is not proved true from earlier propositions. Its purpose is to make precise the idea that the later terms of a sequence become arbitrarily close to one another.

Let $(X,d)$ be a metric space. Here, $X$ is a set of points and $d(x,y)$ is the distance between any two points $x,y\in X$ .

Let $\mathbb{N}=\{1,2,3,\ldots\}$ be the natural numbers.

A sequence $(x_n)_{n\in\mathbb{N}}$ is a function that assigns one point $x_n\in X$ to each natural-number index $n\in\mathbb{N}$ .

Let $\varepsilon>0$ be a positive real number. The value $\varepsilon$ represents a chosen tolerance: it is the greatest separation we are willing to allow between sufficiently late terms of the sequence.

The condition for a Cauchy sequence is:

\forall\varepsilon>0,\ \exists N\in\mathbb{N}\ \text{such that}\ \forall n,m\in\mathbb{N},\quad n,m\geq N \Rightarrow d(x_n,x_m)<\varepsilon.

The symbol $\forall$ means “for every,” and the symbol $\exists$ means “there exists.”

The condition begins with “for every $\varepsilon>0$ .” This means the sequence must meet the requirement for any desired degree of closeness, including extremely small tolerances such as $\varepsilon=0.1$ , $\varepsilon=0.001$ , or $\varepsilon=10^{-100}$ .

For each chosen tolerance $\varepsilon$ , there must be an index $N$ . The index $N$ may depend on $\varepsilon$ : demanding a smaller tolerance may require moving further along the sequence.

The condition $n,m\geq N$ means that both $x_n$ and $x_m$ are selected from the tail of the sequence, beginning at $x_N$ .

The inequality $d(x_n,x_m)<\varepsilon$ means that every pair of terms in this tail lies within the chosen tolerance of each other.

Thus, after some sufficiently late point in the sequence, all remaining terms form a cluster whose members can be made arbitrarily close together.

The definition compares pairs of later terms $x_n$ and $x_m$ rather than comparing each term with a proposed limit. This is useful because it identifies sequences that are approaching a single value without requiring that value to be known in advance.

A Cauchy sequence therefore expresses internal convergence: its later terms increasingly agree with one another. Whether the sequence actually converges to a point in $X$ is a separate question, answered by the completeness of the metric space.

Examples

Constant Sequence
Let $x_n=5$ for every $n\in\mathbb{N}$ .
$x_n=5.$
This is a Cauchy sequence because every pair of terms is identical:
$|x_n-x_m|=|5-5|=0.$
Since $0<\varepsilon$ for every $\varepsilon>0$ , all terms are within every positive tolerance of one another.
Reciprocal Sequence
Let $x_n=\frac{1}{n}$ for every $n\in\mathbb{N}$ .
$x_n=\frac{1}{n}.$
This is a Cauchy sequence because its terms eventually lie arbitrarily close to $0$ . If $n,m\geq N$ , then:
$\left|\frac{1}{n}-\frac{1}{m}\right| < \frac{1}{N}.$
For every $\varepsilon>0$ , choose $N\in\mathbb{N}$ such that $\frac{1}{N}<\varepsilon$ . Then:
$|x_n-x_m|<\varepsilon.$
Geometric Sequence
Let $r$ be a real number satisfying $|r|<1$ , and let:
$x_n=r^n.$
This is a Cauchy sequence because $r^n$ becomes arbitrarily close to $0$ as $n$ increases.
For every $\varepsilon>0$ , there exists $N\in\mathbb{N}$ such that $|r|^N<\frac{\varepsilon}{2}$ . If $n,m\geq N$ , then:
$|x_n-x_m| \leq |x_n|+|x_m| < \frac{\varepsilon}{2}+\frac{\varepsilon}{2} = \varepsilon.$
Rational Approximations of $\sqrt{2}$
Let $X=\mathbb{Q}$ , and let $x_n$ be the decimal approximation of $\sqrt{2}$ truncated after $n$ decimal places:
$x_n = \frac{\lfloor 10^n\sqrt{2}\rfloor}{10^n}.$
This is a Cauchy sequence in $\mathbb{Q}$ because every $x_n$ lies within $10^{-n}$ of $\sqrt{2}$ :
$$
\sqrt{2}-10^{-n}0 $, choose$ N $such that$ 10^{-N}<\varepsilon$. Then:
$|x_n-x_m|<\varepsilon.$
This example is important because the sequence is Cauchy in $\mathbb{Q}$ but does not converge to a rational number.
Natural-Number Sequence — Not Cauchy
Let:
$x_n=n.$
This is not a Cauchy sequence because its later terms do not cluster together.
Choose $\varepsilon=1$ . For every proposed index $N$ , choose:
$n=N \qquad \text{and} \qquad m=N+1.$
Then $n,m\geq N$ , but:
$|x_n-x_m| = |N-(N+1)| = 1.$
Therefore:
$|x_n-x_m|\not<\varepsilon.$
Alternating Sequence — Not Cauchy
Let:
$x_n=(-1)^n.$
This sequence alternates indefinitely between $1$ and $-1$ .
Choose $\varepsilon=1$ . For every proposed index $N$ , there are an even index $n\geq N$ and an odd index $m\geq N$ . Therefore:
$x_n=1 \qquad \text{and} \qquad x_m=-1.$
Thus:
$|x_n-x_m| = |1-(-1)| = 2.$
Since $2\not<1$ , the sequence is not Cauchy.
Square-Root Sequence — Not Cauchy
Let:
$x_n=\sqrt{n}.$
This is not a Cauchy sequence because its terms continue to spread apart.
Choose $\varepsilon=1$ . For every proposed index $N$ , choose:
$n=N \qquad \text{and} \qquad m=4N.$
Then $n,m\geq N$ , and:
$|x_n-x_m| = |\sqrt{N}-\sqrt{4N}| = \sqrt{N}.$
Since $\sqrt{N}\geq1$ , it follows that:
$|x_n-x_m|\not<1.$
Periodic Oscillation — Not Cauchy
Let:
$x_n=\sin\left(\frac{n\pi}{2}\right).$
This sequence repeatedly takes the values $1$ , $0$ , and $-1$ rather than forming an increasingly tight cluster.
Choose $\varepsilon=1$ . For every proposed index $N$ , choose sufficiently large $k\in\mathbb{N}$ such that:
$4k+1\geq N \qquad \text{and} \qquad 4k+3\geq N.$
Then:
$x_{4k+1}=1 \qquad \text{and} \qquad x_{4k+3}=-1.$
Therefore:
$|x_{4k+1}-x_{4k+3}| = |1-(-1)| = 2.$
Since $2\not<1$ , the sequence is not Cauchy.

Evidence

No source evidence recorded.

Related Concepts

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