Cauchy Sequence

A Cauchy sequence is a sequence whose terms become arbitrarily close to one another sufficiently far along the sequence. In mathematical analysis, it provides a way to study convergence without first identifying a candidate limit. The defining condition compares terms with other terms, rather than with a proposed limiting point. A space in which every Cauchy sequence converges to a point belonging to that space is called complete. (live.ocw.mit.edu)

Definition

Let $(X,d)$ be a metric space, and let $(x_n)_{n=1}^{\infty}$ be a sequence of points in $X$. It is Cauchy if

Here $N$ may depend on $\varepsilon$, but the same $N$ must work for every pair of indices $m,n\ge N$. Thus the entire tail of the sequence must fit within arbitrarily small pairwise distances. No condition is imposed on any fixed finite collection of initial terms. (live.ocw.mit.edu)

For sequences of real numbers or complex numbers, the usual distance gives the equivalent condition

In a normed vector space, distance is measured by the norm of the difference, so the condition becomes $\|x_m-x_n\|<\varepsilon$. The concept therefore applies to numbers, vectors, and functions, provided an appropriate metric has been specified. (jirka.org)

Relationship to convergence

Every convergent sequence in a metric space is Cauchy. If $x_n\to x$, choose $N$ so that $d(x_n,x)<\varepsilon/2$ whenever $n\ge N$. The triangle inequality then yields

The converse need not hold: mutual closeness does not ensure that the surrounding space contains the point toward which the sequence approaches. (live.ocw.mit.edu)

A complete metric space is, by definition, one in which every Cauchy sequence converges. Both $\mathbb R$ and $\mathbb C$, and finite-dimensional Euclidean spaces with Euclidean distance, are complete. Consequently, in these spaces the Cauchy criterion characterizes convergence entirely through comparisons between sequence terms. (ocw.mit.edu)

Every Cauchy sequence is bounded, and every subsequence of it is Cauchy. Moreover, if a Cauchy sequence has a convergent subsequence, the entire sequence converges to the same limit. These properties explain why Cauchy conditions are useful in establishing existence of limits, rather than merely testing an already known answer. (jirka.org)

Examples and non-examples

The sequence $x_n=1/n$ is Cauchy in $\mathbb R$. Indeed, for $m,n\ge N$,

which becomes arbitrarily small. It converges to $0$. The same sequence, regarded as taking values in $(0,1]$ with its inherited metric, remains Cauchy but has no limit within that space. Completeness therefore concerns both the distance and the available points. (ocw.mit.edu)

Another example consists of the decimal truncations

of $\sqrt2$. These form a Cauchy sequence of rational numbers, but their real limit is an irrational number. Hence $\mathbb Q$, with its usual metric, is not complete. This illustrates the role of Cauchy sequences in distinguishing the rational and real number systems. (webspace.maths.qmul.ac.uk)

Boundedness alone is insufficient: $(-1)^n$ is bounded but not Cauchy, because arbitrarily late terms can remain distance $2$ apart. Likewise, the condition $|x_{n+1}-x_n|\to0$ is insufficient. The partial sums of the harmonic series have successive differences tending to zero, yet

Thus some pairs of late partial sums remain separated by a fixed positive amount. These examples directly expose the importance of quantifying over all late pairs, not merely neighboring terms. (jirka.org)

Completion of spaces

Cauchy sequences provide a standard construction of the completion of a metric space. Two Cauchy sequences $(x_n)$ and $(y_n)$ are declared equivalent when

This defines an equivalence relation. The distance between their equivalence classes is

The resulting space is complete. The original space embeds by identifying each point with its constant sequence, and its image is a dense subset of the completion. The completion is unique up to a distance-preserving bijection that respects this embedding. (ocw.mit.edu)

Applying this construction to the rationals with ordinary absolute-value distance produces the real numbers. Completing normed spaces produces Banach spaces, while completing inner product spaces in their induced norm produces Hilbert spaces. These constructions are central in functional analysis. (ocw.mit.edu)

Function sequences and iterative methods

For continuous functions on a closed bounded interval, the supremum norm measures the largest difference between two functions. A sequence is Cauchy in this norm precisely when

The uniform Cauchy criterion characterizes uniform convergence for real-valued functions. The space $C([a,b])$ is complete in the supremum norm, so a Cauchy sequence there has a continuous uniform limit. (ocw.mit.edu)

Cauchy estimates also underpin the Banach fixed-point theorem. For a contraction mapping $T$ with contraction factor $0\le q<1$, the iteration $x_{n+1}=T(x_n)$ satisfies, for $m>n$,

The geometric bound proves that the iterates are Cauchy. In a nonempty complete metric space, they therefore converge to the unique fixed point of $T$. (jirka.org)