Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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45 views

Completeness of $\mathcal{l}^1$?

I am a bit confused with how $\mathcal{l}^1$ can be complete. So we know that the sequence space $\mathcal{l}^1$, equipped with $||\cdot||_{\mathcal{l}^1}$ is complete. But the sequence of sequences $$...
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1answer
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I need help answering and understanding this problem about Cauchy Sequences.

Context: I done this problem awhile back and was looking through my notes on it and my answer seems incorrect. Let $(x_n)_{n{\in}\mathbb{N}}$ be a sequence such that $|x_n-x_{n+1}|\;{\le}\;2^{-n}$ ...
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Does square summability need to hold for every vector in an LVS?

In a Linear Vector space I can have an infinite number of vectors, given that I have an orthonormal basis set. If the LVS has infinite dimensionality then, I learnt that it needs to be square summable....
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2answers
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Why do we need Cantor's diagonal method in the following proof? (Total boundedness implies the possession of a Cauchy Sequence)

I am confused about a remark for the following proof. Statement: Let $X$ be a totally bounded metric space. If $(x_n)_{n\geq 1}$ is a sequence in $X$ then it possesses a Cauchy subsequence. Sketch ...
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25 views

Cauchy convergence for an exponential sequence to its subsequence.

Suppose we have the sequence $e_n(q)$ where the term $e_n(q)=\sum_{k=0}^{q}(\frac{q^k}{k!})$. I have that a certain element $x\in B(q,1/n)$ it is an element of a binomial distribution. Now Let $q \in ...
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2answers
120 views

In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

This is cross-posted and answered on MO here. Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ ...
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2answers
49 views

How do we interpret a zero sequence in the context of ideal theory?

We already know and may simply understand the definition of a zero sequence in $\mathbb{Q}$ - it is just a sequence, which converges towards $0$. Given the context of ideal theory, let $R$ be a ring ...
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1answer
52 views

Equivalence of Cauchy Sequences and Cauchy Approximations, HoTT

In HoTT book (Homotopy Type Theory), in order to construct Cauchy reals they introduce the notion of Cauchy approximation, which are defined as : $$x \hspace{0.1cm} \colon \mathbb{Q} \to \mathbb{R} \...
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66 views

Constructive proof that every Cauchy sequence of reals converge

Is there a proof that each Cauchy sequence converges in more constructive flavor, without using Bolzano-Weistrass? Is this valid? For each $k$ there exist $h_k$ $$|x_n-x_m|<2^{-k}, \space \space \...
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1answer
43 views

Convergence of a sequence of measures.

On a measure space $(E,\mathcal{E},\mu)$, let $(\mathcal{F}_n)$ a filtration on $\mathcal{E}$, with $\mathcal{F}_n \uparrow \mathcal{E}$, and let $(\mu_n)$ be a sequence of finite measures defined on $...
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Proving the bounds of the cosine sequence without starting with basic trig identities. [duplicate]

If we look at a unit circle we can see that the values of cosine are between -1 and 1. However is there a particular proof for this fact? I have tried using the Euler's identity to arrive at a proof ...
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1answer
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Proving the bounds of cosine

If we look at a unit circle we can see that the values of cosine are between -1 and 1 however is there a particular proof for this fact? I have tried using the Eulers identity to arrive at a proof ...
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2answers
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Completeness of a metric space $(X,d)$ w.r.t the sup metric

$\mathbf{Question}$: Let $X=\{(x_1,x_2,...): x_i \in \mathbb{R}$ and only finitely many $x_i$'s are non-zero $\}$ and $d:X \times X \to \mathbb{R}$ be a metric on $X$ defined by $d(x,y)=\sup_{i \in \...
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3answers
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Given a Cauchy sequence $(x_n)$ such that $\forall M\in \Bbb{N}$, $\exists k,n\ge M$ such that $x_k<0$ and $x_n>0$. Show that $x_n$ converges to $0$.

Suppose a Cauchy sequence $(x_n)$ is such that for every $M\in \mathbb{N}$, there exists a $k\ge M$ and an $n\geq M$ such that $x_k<0$ and $x_n>0$. How do I show $x_n$ converges to $0$? I have ...
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2answers
31 views

Condition on $(x_n)$ equivalent to $\lim x_n \in U$

Let $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ be a Cauchy sequence of real numbers. Let $U \subseteq R$ be an open set - for simplicity, we can suppose $U$ is an open interval $(a,b)$. Is there a ...
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1answer
35 views

Exercise 24(a) Chapter 3 Baby Rudin Proof Verification

Let $X$ be a metric space. (a) Call two Cauchy sequences $\left\{ p_n \right\}$, $\left\{ q_n \right\}$ in $X$ equivalent if $$ \lim_{n \to \infty} d \left( p_n, q_n \right) = 0.$$ Prove that this is ...
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1answer
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Find a counter-expamle to $\lim_n \limsup_m d(a_n, a_m) =0 \implies (a_n)_n \ \text{is cauchy}.$

Let $(a_n)$ be a sequence in a $d$-metric space. I want to find a counter-example to the statement $$\lim_n \limsup_m d(a_n, a_m) =0 \implies (a_n) \ \text{is Cauchy}.$$ I know that $\lim_n \limsup_m ...
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How to prove that a sequence is Cauchy

How do I show if $x_n = \frac{n^2}{n^2 -1/2}$ is a Cauchy sequence? (using the definition of Cauchy sequence) My attempt: A sequence is Cauchy if $ \forall \epsilon>0$ $ \exists N \in \mathbb N$ $\...
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Why can the $n_{\epsilon}$ of the definitions of convergence and Cauchy sequence be the same in the following proposition?

I have the following proposition proved in my lectures notes, but I think there are a couple of errors and there is one think I don't get: If $p_n$ is a Cauchy sequence in a metric space $(X,d)$, the ...
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Why is a constant sequence a Cauchy sequence?

I think that a constant sequence is a Cauchy sequence because of the definition of Cauchy sequence: The sequence $\{q_n\}$ of rationals is a Cauchy sequence if, for every $\epsilon>0$, where $\...
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A construction in a non Cauchy Sequence

Let $(y_n)$ be a sequence which is not Cauchy. Let $\lim_{n\to\infty} d(y_n,y_{n+1})=0$ and $d(y_n,y_{n+1})$ is a decreasing sequence. Then there exists an $\epsilon>0$ such that for every $n\in \...
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Is saying “the sequence converges” the same as saying “the sequence has a limit”?

I'm reading a book on set theory and it has some real analyisis in it, and since i've only studied analysis in college and i didn't understand much honestly now studying by myself i'm struggling a ...
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Completion of surreal numbers

Surreal number field $\mathbf{No}$ is not complete, there are "gaps". Does there exists a completion of it? I know this question depends on axioms of set theory and more, feel free to ...
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1answer
27 views

True / false question about a Cauchy sequence in Real Analysis [duplicate]

I am solving assignments in Real analysis but I am unable to think about how I can solve this question. Let $f: ( 0, \infty ) \to \mathbb{R}$ be a continuous function. Does $f$ maps any Cauchy ...
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1answer
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Backwards direction of Cauchy Criterion for Sequences of Functions

I am reviewing the proof of the Cauchy Criterion for sequences of functions and have a question regarding the backwards direction. Statement: Let $A\subseteq \mathbb{R}$ and $(f_n)$ be a sequence of ...
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Prove: $f:M\to M'$ preserves Cauchy-sequences iff $f$ is extendable to a continuous function $f:c(M)\to c(M')$ (completion)

Let $c(M)$ denote the completion of $M$. Prove: $f:M\to M'$ preserves Cauchy-sequences iff $f$ is extendable to a continuous function $f:c(M)\to c(M')$ My attempt: $\Leftarrow$: Consider a Cauchy-...
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If $\lim_{n \to \infty} (a_{n + 1} - a_n) \to 0$ then $a_n$ converges?

Someone recently tried to claim that $\lim_{n \to \infty} (a_{n + 1} - a_n) = 0$ implies $\lim_{n \to \infty} a_n$ exists. This is of course not true, as $a_n = \log n$ shows. In fact, even if $a_n$ ...
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1answer
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Every Cauchy sequence in $A$ converges in $X$, where $A$ is dense. Show that $X$ is complete.

My question is: Let $( X, d)$ be a metric space and $A$ a dense subset of $X$ such that every Cauchy sequence in $A$ converges in $X$. Prove that $( X, d)$ is complete. Solution: Case 1: If $X = A$ ...
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38 views

Is this proof that Cauchy sequences converge right?

I am trying to prove that a Cauchy sequence must converge. To do this I used the fact that a sequence converges if and only if its lim inf equals its lim sup, as follows: Consider a Cauchy sequence $\...
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1answer
30 views

Every solution $x(t)$ of $x′(t)=A(t)x(t)$ converges, question about the answer

So in the post: Show that every solution $x(t)$ of $x'(t)= A(t)x(t)$ converges to some limit (Long-time asymptotics). Suppose $$\int_0^∞\|A(t)\|\,dt < ∞.$$ Show that every solution $x(t)$ of ...
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1answer
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Multivariable Calculus proof explanation help, Cauchy Sequences

Proof here. This is a proof from the Advanced Calculus book by Gerald.B Folland. I understand all the steps except where the author goes on to say that $|\textbf{x}_k|<|\textbf{x}_{K+1}|+1$ for all ...
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0answers
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Why don't all sequences on compact metric spaces converge?

My thinking is this: Let $(X, d)$ be a compact metric space, thus it is also sequentially compact, meaning that all sequences have a convergent subsequence. We also know that every compact metric ...
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3answers
52 views

let $<a_n>$ be a sequence of real numbers such that $\sum|a_n-a_{n-1}|$ is convergent series. Show that $\sum a_n x^n$ converge on $(-1,1)$

let $<a_n>$ be a sequence of real numbers such that $\sum|a_n-a_{n-1}|$ is convergent series. Show that power series $\sum a_n x^n$ converge on interval $(-1,1)$ How to approach . let $0<\...
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Show if $\,\alpha_n\searrow0$ the sequence $a_n:=\alpha_1-\alpha_2+\alpha_3-\alpha_4+…+(-1)^{n-1}\alpha_n$ converges.

Show if $\,\alpha_n\searrow0$ the sequence $a_n:=\alpha_1-\alpha_2+\alpha_3-\alpha_4+...+(-1)^{n-1}\alpha_n$ converges. $\,\alpha_n\searrow0$ means $\forall n \in \mathbb{N}:\alpha_n\ge\alpha_{n+1}&...
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1answer
60 views

Prove if $\vert a_{n+1}-a_n\vert\leq \ q\vert a_n - a_{n-1} \vert$ then $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence

I’m tasked with proving the following: For $n_0 \in \mathbb{N}$ and $q\in \mathbb{R}$ with $0<q<1$ and $n \geq n_0$, let $(a_n)_{n\in\mathbb{N}}$ be a sequence in $\mathbb{R}$. If $$\vert ...
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Cauchy sequence to prove Integral does not exist

I'm learning Cauchy sequence right now and didn't succeed. prove that: $$ \int_{i=1}^\infty \cfrac{x\,dx}{\sqrt{1+3x+x^2}} $$ Does not exist Using only Cauchy sequence! Thx!
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1answer
52 views

Show that $a_n:=\frac{(-1)^{n-1}}{2n-1}$ converges

Show that $a_n:=\frac{(-1)^{n-1}}{2n-1}$ converges. If $(a_n)$ converges, the sequences is a Cauchy sequence. Which means: $\forall \epsilon >0 \,\,\,\exists N \in \mathbb{N}\,\,\,m,n>N:\left|\...
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0answers
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A proof that $L^{\infty}$ is complete

I know this has been proved in other links, but I am wondering about the validity of the following proof: Suppose $X_n$ is a Cauchy sequence in $L^\infty$. Then there exists a subsequence $Y_k \...
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2answers
53 views

Prove that ln(n) is not a Cauchy Sequence? [duplicate]

Show from the definition of a Cauchy Sequence that Xn=ln(n) is not a Cauchy Series
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1answer
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Cauchy sequence is bounded? (Do we need any element in a sequence to be finite?)

This question is related to the question Is every cauchy sequence bounded? The sequence $\{a_n\}$ used in that question $$a_n=\frac{1}{n-1}$$ has the first element $a_1\rightarrow\infty$...
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3answers
36 views

Show that $f$ can be extended to a Cauchy-sequence preserving continuous mapping on $\overline{A}$.

Question: Let $(X,d)$ be a metric space and $A\subset X.$ If $f: A\to\mathbb R$ be a Cauchy-sequence preserving continuous mapping then show that $f$ can be extended to a Cauchy-sequence preserving ...
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27 views

Prove Cauchy criterion without using the bounded property

The Cauchy criterion states that: A sequence is Cauchy if and only if it converges. Almost all proofs I can find for this theorem use the fact that: ...
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1answer
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Cauchy sequences in $\mathbb{C}$ does not imply that the function converges.

I am currently working on a proof and I have the current properties: A Cauchy sequence $\{ f_n\}_{n\in \mathbb{N}} $ of functions in some space $X$ with the property that $|(f_n - f_m)(x)|<\...
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2answers
79 views

How do I show that if $p_n q_{n-1} - p_{n-1}q_n = 1$, then $p_n/q_n$ converges?

Let $p_{n}$ and $q_{n}$ be strictly increasing, integer valued, sequences. Show that if $$p_{n}q_{n-1}-p_{n-1}q_{n}=1,$$ for each integer $n \ge 1$, then the sequence of quotients $\frac{p_{n}}{q_{n}}$...
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0answers
51 views

Show that every Cauchy sequence which admits a convergent subsequence is also convergent. Is my proof correct?

Let $(x_{n})_{n=m}^{\infty}$ be a Cauchy sequence in $(X,d)$. Suppose that there is some subsequence $(x_{f(n)})_{n=m}^{\infty}$ of this sequence which converges to a limit $x_{0}\in X$. Then the ...
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2answers
35 views

Basic Cauchy sequence convergence question

Let $f: ]0,1[ \to \mathbb{R}$ be a function. Suppose that for every sequence $(\epsilon_n)_n$ in $]0,1[$ with $\epsilon_n \searrow 0$ we have that $(f(\epsilon_n))_n$ is a Cauchy sequence. Can we ...
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0answers
23 views

Proving that Cauchy Summation converges [duplicate]

How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$ I tried to prove that by proving that the following summation in in ...
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2answers
79 views

Cauchy product summation converges

I had a previous question here, which I'm quoting: How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1) (n-k+1)}$$ I tried to prove ...
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0answers
67 views

Prove that absolute convergence of sup norms implies uniform convergence of functions (Weierstrass M-test)

Let $(X,d)$ be a metric space, and let $f_{n}$ be a sequence of bounded real-valued continuous functions on $X$ such that the series $\sum_{n=1}^{\infty}\|f_{n}\|$ is convergent. Then the series $\...
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1answer
19 views

Rigor behind choice of index for Cauchy squence in bounded function space.

We have a Cauchy sequence in the normed space of bounded functions $(f_n)_n \subset B(\Omega, \mathbb{K})$. I have shown that $(f_n(\omega))_n$ is a Cauchy squence for all $\omega \in \Omega$. What's ...

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