What is Cauchy sequences: Definition and 49 Discussions

In mathematics, a Cauchy sequence (French pronunciation: [koʃi]; English: KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.
It is not sufficient for each term to become arbitrarily close to the preceding term. For instance, in the sequence of square roots of natural numbers:






{\displaystyle a_{n}={\sqrt {n}},}
the consecutive terms become arbitrarily close to each other:


















{\displaystyle a_{n+1}-a_{n}={\sqrt {n+1}}-{\sqrt {n}}={\frac {1}{{\sqrt {n+1}}+{\sqrt {n}}}}<{\frac {1}{2{\sqrt {n}}}}.}
However, with growing values of the index n, the terms an become arbitrarily large. So, for any index n and distance d, there exists an index m big enough such that am – an > d. (Actually, any m > (√n + d)2 suffices.) As a result, despite how far one goes, the remaining terms of the sequence never get close to each other, hence the sequence is not Cauchy.
The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself, as opposed to the definition of convergence, which uses the limit value as well as the terms. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination.
Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters and Cauchy nets.

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  1. H

    If ##|s_{n+1} - s_n| \lt 1/2^n##, then ##(s_n)## is a Cauchy sequence

    My attempt: It can be proved that ##\lim \frac{1}{2^n} = 0##. Consider, ##\frac{\varepsilon}{k} \gt 0##, there exists ##N##, such that $$ n \gt N \implies \frac{1}{2^n} \lt \varepsilon $$ Take any ##m,n \gt N##, and such that ##m - k = n##. ##|s_m - s_{m-1} | \lt \frac{1}{2^{m-1}} \lt...
  2. B

    Do Cauchy Sequences Imply Convergent Differences?

    I've started by writing down the definitions, so we have $$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$ $$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w...
  3. yucheng

    Understanding the Use of Min in Cauchy Sequences

    I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ I am having trouble understanding the purpose / motivation behind using the min as in ##\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)## and ##\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}...
  4. yucheng

    Proof that two equivalent sequences are both Cauchy sequences

    Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##. A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##. We proceeded by...
  5. L

    A Same open sets + same bounded sets => same Cauchy sequences?

    Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...
  6. Euler2718

    Showing a sequence of functions is Cauchy/not Cauchy in L1

    Homework Statement Determine whether or not the following sequences of real valued functions are Cauchy in L^{1}[0,1]: (a) f_{n}(x) = \begin{cases} \frac{1}{\sqrt{x}} & , \frac{1}{n+1}\leq x \leq 1 \\ 0 & , \text{ otherwise } \end{cases} (b) f_{n}(x) = \begin{cases} \frac{1}{x} & ...
  7. T

    Metric space of continuous & bounded functions is complete?

    Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
  8. T

    I Proof Explanation: Showing an extension to a continuous function

    I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##...
  9. T

    Regarding Real numbers as limits of Cauchy sequences

    Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...
  10. Math Amateur

    MHB Cauchy Sequences and Completeness in R^n ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Theorem 1.6.5 (Completeness of \mathbb{R}^n) ... Duistermaat and Kolk"s Theorem 1.6.5 and its...
  11. C

    Do Cauchy sequences always converge?

    Hello evry body let be $(u_{n}) \in \mathbb{C}^{\matbb{N}}$ with $u_{n}^{2} \rightarrow 1$ and $\forall n \in \mathbb{N} (u_{n+1) - u_{n}) < 1$. Why does this sequences converge please? Thank you in advance and have a nice afternoon:oldbiggrin:.
  12. Math Amateur

    MHB Apostal Chapter 4 - Cauchy Sequences - Example 1, Section 4.3, page 73

    I need some help in fully understanding Example 1, section 4.3 Cauchy Sequences, page 73 of Apostol, Mathematical Analysis. Example 1, page 73 reads as follows: https://www.physicsforums.com/attachments/3844 https://www.physicsforums.com/attachments/3845 In the above text, Apostol writes: "...
  13. evinda

    MHB Cauchy Sequences: What it Means to be $|x_{n+1}-x_n|_p< \epsilon$

    Hi! (Wave) I am looking at the following exercise: If $\{ x_n \}$ is a sequence of rationals, then this is a Cauchy sequence as for the p-norm, $| \cdot |_p$, if and only if : $$\lim_{n \to +\infty} |x_{n+1}-x_n|_p=0$$ That's what I have tried: $\lim_{n \to +\infty} |x_{n+1}-x_n|_p=0$ means...
  14. Fredrik

    Cauchy sequences and absolutely convergent series

    Homework Statement I want to prove that if X is a normed space, the following statements are equivalent. (a) Every Cauchy sequence in X is convergent. (b) Every absolutely convergent series in X is convergent. I'm having difficulties with the implication (b) ⇒ (a). Homework Equations Only...
  15. D

    Are there any metric spaces with no Cauchy sequences?

    A metric space is considered complete if all Cauchy sequences converge within the metric space. I was just curious if you could have a case of a metric space that doesn't have any Cauchy sequences in it. Wouldn't it be complete by default? When trying to think of a space with no cauchy...
  16. A

    MHB An Equivalence Relation with Cauchy Sequences

    We let C be the set of Cauchy sequences in \mathbb{Q} and define a relation \sim on C by (x_i) \sim (y_i) if and only if \lim_{n\to \infty}|x_n - y_n| = 0. Show that \sim is an equivalence relation on C. We were given a hint to use subsequences, but I don't think they are really necessary...
  17. alyafey22

    MHB Complete spaces and Cauchy sequences

    I know that a metric space is complete if every Cauchy sequence converges that will surely designate compact metric spaces as complete spaces . I need to see examples of metric spaces which are not complete. Thanks in advance !
  18. A

    The difference between the limits of two Cauchy Sequences

    Lets say that we have two Cauchy sequences {fi} and {gi} such that the sequence {fi} converges to a limit F and the sequence {gi} converges to a limit G. Then it can easily be shown that the sequence defined by { d(fi, gi) } is also Cauchy. But is it true that this sequence, { d(fi, gi) }...
  19. S

    Cauchy Sequences and Convergence

    Homework Statement Prove the following theorem, originally due to Cauchy. Suppose that (a_{n})\rightarrow a. Then the sequence (b_{n}) defined by b_{n}=\frac{(a_{1}+a_{2}+...+a_{n})}{n} is convergent and (b_{n})\rightarrow a. Homework Equations A sequence (a_{n}) has the Cauchy property...
  20. S

    Cauchy sequences is my proof correct?

    Homework Statement Let (xn)n\inℕ and (yn)n\inℕ be Cauchy sequences of real numbers. Show, without using the Cauchy Criterion, that if zn=xn+yn, then (zn)n\inℕ is a Cauchy sequence of real numbers. Homework Equations The Attempt at a Solution Here's my attempt at a proof: Let...
  21. M

    Prove: Cauchy sequences are converging sequences

    Homework Statement I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence Homework Equations What I know is: a[n] is bounded any subsequence is bounded there exists a monotone subsequence all monotone bounded sequences converge there exists a...
  22. F

    Proof of "Every Cauchy Sequence is Bounded

    I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded I don't understand that how we can take m=N_{1} while m>N_{1} ? In fact i mean that in a metric space (A,d) can we say that...
  23. I

    Why Continuous Functions Don't Preserve Cauchy Sequences

    Homework Statement Why is it that continuous functions do not necessarily preserve cauchy sequences. Homework Equations Epsilon delta definition of continuity Sequential Characterisation of continuity The Attempt at a Solution I can't see why the proof that uniformly continuous...
  24. matqkks

    Why are Cauchy sequences important in understanding limits and completeness?

    Why are Cauchy sequences important? Is there only purpose to test convergence of sequences or do they have other applications? Is there anything tangible about Cauchy sequences
  25. R

    Cauchy sequences and continuity versus uniform continuity

    Homework Statement This isn't really a problem but it is just something I am curious about, I found a theorem stating that you have two metric spaces and f:X --> Y is uniform continuous and (xn) is a cauchy sequence in X then f(xn) is a cauchy sequence in Y. Homework Equations This...
  26. B

    Can Cauchy Sequences be Bounded? Theorem 1.4 in Introduction to Analysis

    Homework Statement Theorem 1.4: Show that every Cauchy sequence is bounded. Homework Equations Theorem 1.2: If a_n is a convergent sequence, then a_n is bounded. Theorem 1.3: a_n is a Cauchy sequence \iff a_n is a convergent sequence. The Attempt at a Solution By Theorem 1.3, a...
  27. M

    Cauchy Sequences - Complex Analysis

    Hope someone could give me some help with a couple of problems. First: Proof of - A function f:G -->Complex Plane is continuous on G iff for every sequence C(going from 1 to infinity) of complex numbers in G that has a limit in G we have limit as n --> infinity f(C) = f(limit as n...
  28. E

    Cauchy Sequences in General Topological Spaces

    "Cauchy" Sequences in General Topological Spaces Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every...
  29. M

    Uniform continuity, cauchy sequences

    Homework Statement If f:S->Rm is uniformly continuous on S, and {xk} is Cauchy in S show that {f(xk)} is also cauchy. Homework Equations The Attempt at a Solution Since f is uniformly continuous, \forall\epsilon>0, \exists\delta>0: \forallx, y ∈ S, |x-y| < \delta =>...
  30. Fredrik

    Equivalence classes of Cauchy sequences

    \mathbb R can be defined as "any (Dedekind-)complete ordered field". This type of abstract definition is a different kind than e.g. the "equivalence classes of Cauchy sequences" construction. I prefer abstract definitions over explicit constructions, so I would be interested in seeing similar...
  31. M

    Cauchy sequences, induction, telescoping property

    Homework Statement Scanned and attached Homework Equations I am guessing a combination of induction and the telescoping property. The Attempt at a Solution I'm studying this extramurally, and I've just hit a wall with this last chunk of the sequences section, so if someone can...
  32. D

    Proving Cauchy Sequences with Cosine Function

    Homework Statement Well, my problem is proving that sequences are in fact Cauchy sequences. I know all the conditions that need to be satisfied yet I cannot seem to apply it to questions. (Well, only the easy ones!) My question is, prove that X_{n} is a Cauchy sequence, given that...
  33. G

    Completeness of R2 with Taxicab Norm

    Homework Statement Given R is complete, prove that R2 is complete with the taxicab norm The Attempt at a Solution you know that ,xk \rightarrow x , yk \rightarrow y Then, given \epsilon, choose Nx and Ny so that \left|x_n - x_m\left| and \left|y_n - y_m\left| are less than...
  34. J

    Cauchy sequences and uniform convergence

    Homework Statement Suppose the infinite series \sum a_v is NOT absolutely convergent. Suppose it also has an infinite amount of positive and an infinite amount of negative terms. Homework Equations The Attempt at a Solution Say we want to prove it converges by proving...
  35. J

    What is the mistake in my reasoning for Cauchy sequences?

    As far as I understand, a sequence converges if and only if it is Cauchy. So say for some sequence a_n and for all epsilon greater than zero we have |a_n - a_{n+1}| < \epsilon for large enough n. We could then say a_n converges if and only if \lim_{n \rightarrow \infty} a_n - a_{n+1} = 0 ...
  36. C

    Proving Cauchy Sequences: Infinite Subsequences

    Hi, I need to prove that any infinite subsequence {xnk}of a Cauchy sequence {xn}is a Cauchy sequence equivalent to {xn}. My problem is that it seemed way too easy, so I'm concerned that I missed something. Please see the attachment for my solution, and let me know what you think. Thanks.
  37. E

    Cauchy Sequences Triangle Inequality.

    Homework Statement assuming an and bn are cauchy, use a triangle inequality argument to show that cn= | an-bn| is cauchy Homework Equations an is cauchy iff for all e>0, there is some natural N, m,n>=N-->|an-am|<e The Attempt at a Solution I am currently trying to work backwards on this one...
  38. I

    Sequences limits and cauchy sequences

    Homework Statement prove or refute: if lim(a(2n)-a(n)=o , then a(n) is a cauchy sequence Homework Equations The Attempt at a Solution I need to prove that for every m,n big enough a(m)-a(n)<epsilon so I know for all m and n I can say m=l*n, lim(a(m)-a(n))=lim(a(n*l)-a(n*l/2)...
  39. J

    Cauchy sequences and sequences in general

    Is every sequence that converges a Cauchy sequence (in that for every e > 0, there is an integer N such that |a_n - a_m| < e whenever n,m > N)? I think it is because if a sequence a_n converges to L, then you can mark off an open interval of any size about L such that this interval contains all...
  40. G

    Cauchy Sequences: Definition & a(m) Clarification

    By definition, a sequence a(n) has the Cauchy sequence if for eery E>0 ,there exist a natural number N such that Abs(a(n) - a(m) ) < E for all n, m > N Could anyone tell me what is a(m) ? is it a subsequence of a(n) , or could it be any other non related sequence ?
  41. D

    Understanding Cauchy Sequences in Banach Spaces

    Homework Statement http://img394.imageshack.us/img394/5994/67110701dt0.png Homework Equations A banach space is a complete normed space which means that every Cauchy sequence converges. The Attempt at a Solution I'm stuck at exercise (c). Suppose (f_n)_n is a Cauchy sequence in E. Then...
  42. P

    Divergent Harmonic Series, Convergent P-Series (Cauchy sequences)

    Homework Statement (a) Show that \sum \frac 1n is not convergent by showing that the partial sums are not a Cauchy sequence (b) Show that \sum \frac 1{n^2} is convergent by showing that the partial sums form a Cauchy sequenceHomework Equations Given epsilon>0, a sequence is Cauchy if there...
  43. R

    Proving Cauchy Sequence: a_n = [a_(n-1) + a_(n-2)]/2

    Homework Statement Prove that the following sequence is Cauchy: a_n = [a_(n-1) + a_(n-2)]/2 (i.e. the average of the last two), where a_0 = x a_1 = y Homework Equations None The Attempt at a Solution I was trying to use the definition of Cauchy (i.e. |a_m - a_n| < e) by...
  44. R

    Solving Problems in a Class Missed: Limits, Isolated Points & Cauchy Sequences

    This is review from class the other day that I managed to miss because of illness and I was wondering if someone could explain how to go about solving these problems: #1 Let B = \left\{ \frac{(-1)^nn}{n+1}:n = 1,2,3,...\right\} Find the limit points of B Is B a closed set? Is B an open set...
  45. D

    Proving Cauchy Sequences with Totient Theorem

    Homework Statement If p does not divide a, show that a_n=a^{p^{n}} is Cauchy in \mathbb{Q}_p. The Attempt at a Solution We can factor a^{p^{n+k}}-a^{p^n}=a^{p^n}(a^{p^{n+k}-1}-1). p doesn't divide a^{p^n} so somehow I must show that a^{p^{n+k}-1}-1 is divisible by larger and larger powers of...
  46. Oxymoron

    Proving Cauchy Sequences in the p-adic Metric

    Question Consider the sequence \{p^n\}_{n\in\mathbb{N}}. Prove that this sequence is Cauchy with respect to the p-adic metric on \mathbb{Q}. What is the limit of the sequence?
  47. S

    Cauchy Sequence: Understanding the Boundary Condition

    hello all I found this rather interesting suppose that a sequence {x_{n}} satisfies |x_{n+1}-x_{n}|<\frac{1}{n+1} \forall n\epsilon N how couldn't the sequence {x_{n}} not be cauchy? I tried to think of some examples to disprove it but i didnt achieve anything doing that, please...
  48. Oxymoron

    Cauchy sequences in an inner product space

    Im in need of some guidance. No answers, just guidance. :smile: Question. Let (x_m) be a Cauchy sequence in an inner product space, show that \left\{\|x_n\|:n=1,\dots,\infty\right\} is bounded. proof From the definition we know that all convergent sequences are Cauchy...
  49. T

    Prove Cauchy Sequence: {sn} from {tn}

    Let {an}(n goes from 1 to infinity) be a sequence. For each n define: sn=Summation(j=1 to n) of aj tn=Summation(j=1 to n) of the absolute value of aj. Prove that if {tn}(n goes from 1 to infinity) is a Cauchy sequence, then so is {sn}(n goes from 1 to infinity). I started this...