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:
a
n
=
n
,
{\displaystyle a_{n}={\sqrt {n}},}
the consecutive terms become arbitrarily close to each other:
a
n
+
1
−
a
n
=
n
+
1
−
n
=
1
n
+
1
+
n
<
1
2
n
.
{\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.
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...
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...
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}...
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...
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##...
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} & ...
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...
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##...
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...
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...
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:.
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:
"...
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...
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...
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...
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...
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 !
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) }...
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...
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...
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...
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...
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...
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
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...
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...
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...
"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...
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 =>...
\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...
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...
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...
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...
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...
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 ...
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.
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...
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)...
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...
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 ?
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...
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...
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...
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...
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...
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?
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...
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...
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...