Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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66
votes
5answers
11k views

Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
26
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2answers
25k views

Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges ...
19
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5answers
2k views

Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy? I know that Cauchy condition means that for each $\varepsilon>0$ there exists $N$ such that $d(x_p,x_q)<\...
5
votes
2answers
148 views

Show that the sequence $a_1=1$, $a_2=2$, $a_{n+2} = (a_{n+1}+a_n)/2$ converges by showing it is Cauchy

Show that the sequence $a_1=1$, $a_2=2$, $a_{n+2} = (a_{n+1}+a_n)/2$ converges by showing it is Cauchy. My work : Need to show that for every $\epsilon \gt 0$ there exist $N$ such that $n,m\ge N \...
19
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3answers
32k views

Proving that a sequence such that $|a_{n+1} - a_n| \le 2^{-n}$ is Cauchy

Suppose the terms of the sequence of real numbers $\{a_n\}$ satisfy $|a_{n+1} - a_n| \le 2^{-n}$ for all $n$. Prove that $\{a_n\}$ is Cauchy. My Work So by the definition of a Cauchy sequence, for ...
17
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2answers
4k views

Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$. Is $f$ continuous? Let $f$ be ...
6
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4answers
3k views

Pseudo-Cauchy sequence

I have never seen this terminology before, so I will provide the given definition. A Pseudo-Cauchy sequence is : A sequence $(a_n)$ if for any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such ...
4
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3answers
679 views

Cauchy Sequence. What is this question actually telling me?

Let $(a_n)$ be a sequence such that $\lim\limits_{N\to\infty} \sum_{n=1}^n |a_n-a_{n+1}|<\infty$. Show that $(a_n)$ is Cauchy. So basically I am told that the sum of the difference isn't infinite. ...
5
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2answers
5k views

Metric space is totally bounded iff every sequence has Cauchy subsequence

Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence. I think I proved the Cauchy subsequence part: Let $a_{0},a_{1}, a_{2}, a_{3}, a_{4},...\in X$ be ...
3
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5answers
1k views

If $\{x_n\}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is $\{x_n\}$ a Cauchy sequence?

Since the definition of Cauchy sequence is: Understanding the definition of Cauchy sequence, I noticed we need an absolute value for $a_m-a_n$ in the definition so the statement would be false. But I ...
15
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1answer
6k views

If a subsequence of a Cauchy sequence converges, then the whole sequence converges.

Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon > 0$. Take $N >...
5
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3answers
2k views

A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy

Give an example of a sequence $(x_n)$ of real numbers, where $\displaystyle\lim_{n\to+\infty}|x_n-x_{n+1}|=0$, but $(x_n)$ is not a Cauchy sequence
3
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2answers
887 views

Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?

If I have a sequence {$a_n$} that has the property of $\lim(a_{n+1}-a_n)=0$, does that make it a Cauchy Sequence. I think it doesn't because $a_n = \sum_{k=1}^n \frac{1}{k}$ is a counter example. ...
5
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1answer
2k views

Banach spaces and their unit sphere

Let $X$ be a normed vector space. Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete if ...
8
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1answer
1k views

Which of the following metric spaces are complete?

[NBHM_2006_PhD Screening test_Topology] Which of the following metric spaces are complete? $X_1=(0,1), d(x,y)=|\tan x-\tan y|$ $X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$ $X_3=\mathbb{Q}, ...
2
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1answer
694 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \ge 0$. Proof: I will attempt to show that the ...
25
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4answers
21k views

Cauchy sequence is convergent iff it has a convergent subsequence

Prove that if $\left ( x_{n} \right )$ is a Cauchy sequence in a metric space X then $\left ( x_{n} \right )$ is convergent if and only if $\left ( x_{n} \right )$ has a convergent subsequence. Note: ...
13
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1answer
12k views

How do I prove a uniformly continuous function preserves Cauchy sequences?

Let $f$ be a uniformly continuous function on A of $\Bbb{R}$. How do I show that if $a_n$ is Cauchy, then $f(a_n)$ is Cauchy. This is what I have worked on, but it does not quite make sense since I ...
10
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2answers
3k views

Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$

Let $E \subset F \subset X$ and $f:E\rightarrow Y$. We say that the function $g:F\rightarrow Y$ is an extension of $f$ if $g(x) = f(x)$ for all $x \in E$. Let $f: (a, b) \rightarrow \mathbb{R}$. ...
23
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2answers
6k views

A “non-trivial” example of a Cauchy sequence that does not converge?

A Cauchy sequence doesn't necessarily converge, e.g. take the sequence $(1/n)$ in the space $(0,1)$. Maybe my intuition is wrong but I tend to think of this as, "it does converge but what it ...
6
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2answers
4k views

Show that every monotonic increasing and bounded sequence is Cauchy.

The title is kind of misleading because the task actually to show Every monotonic increasing and bounded sequence $(x_n)_{n\in\mathbb{N}}$ is Cauchy without knowing that: Every bounded non-empty ...
1
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3answers
4k views

Please help prove that $(x_n)$ is a Cauchy sequence if $|x_{n+1} - x_n| \leq Cr^n$

Past final exam question for an intro to Real Analysis course: Let $C > 0$, $0<r<1$ and suppose that $\forall n\in \mathbb N, |x_{n+1} - x_n| \leq Cr^n$. Please help me prove that $(x_n)$ is ...
3
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1answer
1k views

$C[0,1]$ with $L^1$ norm is not Banach space.

I want to check that $(C[0,1],∥⋅∥_1)$ is not a Banach space, where $\|f\|_1 = \int_0^1 |f(x)|\,{\rm d}x$.I took $(f_n)_{n \geq 1}$ a sequence in $C[0,1]$ given by:$f_n: [0,1] \rightarrow \mathbb{R}, \...
10
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3answers
1k views

Cauchy Sequence in $X$ on $[0,1]$ with norm $\int_{0}^{1} |x(t)|dt$

In Luenberger's Optimization book pg. 34 an example says "Let $X$ be the space of continuous functions on $[0,1]$ with norm defined as $\|x\| = \int_{0}^{1} |x(t)|dt$". In order to prove $X$ is ...
1
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1answer
388 views

Proving that Euclidean Space is a complete metric space

I start my proof with supposing that I have a Cauchy sequence $x^\Bbb{R^n}_k$ in $\Bbb{R^n}$- that is, $x^\Bbb{R^n}_k=(x_1, x_2,...,x_N,...,x_m,..,x_n...)$ ,where $x_i=(x^i_1,x^i_2,x^i_3,...,x^i_n)$ e....
9
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1answer
713 views

Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then

(UGC CSIR-2015, DECEMEMBER, MATHEMATICAL SCIENCES) $f$ maps a bounded sequence to a bounded sequence. $f$ maps a Cauchy sequence to a Cauchy sequence. $f$ maps a convergent sequence to a ...
3
votes
1answer
926 views

Converge uniformly on open interval implies on closed interval

Suppose $f_n(x)$ is defined on $[a,b]$, and it converges uniformly to $f(x)$ on $(a,b)$. And the sequences $f_n(a)$ and $f_n(b)$ both converge (say, to points $c$ and $d$ respectively). I want to ...
5
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5answers
1k views

I would like to know an intuitive way to understand a Cauchy sequence and the Cauchy criterion.

My understanding from the definition in my book (Rudin) is this. A seq. $\{p_n\}$ in a metric space $X$ (I only really know $\mathbb R^k$) is said to be a Cauchy sequence if for any given $\epsilon ...
6
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1answer
3k views

Completeness of the set of convergent sequences

It's a problem from the book "Topology of Metric Spaces", written by Kumaresan: "Show that the set $\textbf{c}$ of convergent sequences in the Normed Linear Space of all bounded real sequences ...
2
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1answer
681 views

A Cauchy sequence has a rapidly Cauchy subsequence

I am trying to fill in the details of a proof related to the Riesz-Fischer Theorem. We need to show that every Cauchy sequence $\{f_n\}$ has a rapidly Cauchy subsequence. My text claims that we can ...
2
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2answers
982 views

Show that a Cauchy sequence has a fast-Cauchy subsequence

A sequence $\{x_j\}$ is said to be fast-Cauchy if $\sum_1^\infty d(x_j,x_{j+1})<+\infty$. Show that every Cauchy sequence has a fast-Cauchy subsequence. **My attempt:**Argue by contradiction, ...
2
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3answers
149 views

I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy [duplicate]

I want an example of a sequence that satisfies $\mid x(n) - x(n-1)\mid \to 0$ but not Cauchy ? I tried to find such sequence $x(n)=1/2,1/3,1/2,1/3,1/4,1/2,1/3,1/4,1/5,,,,$ it's not Cauchy since it is ...
1
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3answers
599 views

How to prove the sequence $x_{n+1} = \frac{x_n}2 + \frac 1{x_n}$ is a Cauchy sequence

Here is a question I do not know how to prove. Thanks for your helping! Prove that the sequence $$x_{n+1} = \dfrac{x_n}2 + \dfrac 1{x_n}, x_0 = 1$$is a Cauchy sequence.
2
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3answers
386 views

Prove a sequence is Cauchy and find its limit

Let $a\leq b \in \mathbb{R}$. Show that the sequence $a_1 = a, a_2=b$ and $a_{n+2}=\frac{a_{n+1}+a_n}{2}$ for $n\geq 1$ is Cauchy and find it's limit. I did for $n>m$: $$|a_n-a_m|=\frac{1}{2}|(a_{...
1
vote
3answers
221 views

Prove that the sequence $a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$ is convergent and find its limit

Let $c>0$, $a_{1} = 1$, and $$a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$$ I need to: Show that $a_{n}$ is defined for every $n\geq 1$ Show that this sequence is convergent. Find its ...
7
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1answer
4k views

Continuous Functions and Cauchy Sequences

We know that if a function $f: A \mapsto \mathbb{R}$, $A \subseteq \mathbb{R}$, is uniformly continuous on $A$ then, if $(x_n)$ is a Cauchy sequence in $A$, then $(f(x_n))$ is also a Cauchy sequence. ...
6
votes
1answer
469 views

Prob. 11, Chap. 4 in Baby Rudin: uniformly continuous extension from a dense subset to the entire space

Here is Prob. 11, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $...
3
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3answers
10k views

Confused with proof that all Cauchy sequences of real numbers converge.

First the textbook proves that all Cauchy sequences are bounded, and so have a convergent subsequence, $\{a_{n_{k}}\}$ that converges to a limit, say $L$. Now we use this to prove that all Cauchy ...
3
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2answers
1k views

Continuity, uniform continuity and preservation of Cauchy sequences in metric spaces.

Let $ X $ and $ Y $ be metric spaces, and let $ f: X \to Y $ be a mapping. Determine which of the following statements is/are true. a. If $ f $ is uniformly continuous, then the image of every Cauchy ...
2
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3answers
174 views

Construct such $d$ that $(\mathbb{R} \setminus \mathbb{Z}, d)$ is complete metric space

Good evening! I had topology exam yesterday and I had a question that gave me problems. Lets consider $\mathrm{G} = \mathbb{R} \setminus \mathbb{Z} $ , i.e. real line without integrals, where $\rho$ ...
2
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1answer
3k views

Equivalent Cauchy sequences.

Hi everyone I'm having a bad time with two questions in the Analysis book of Terry Tao. I finally finished one of the exercises and I'm wondering if the next reasoning is correct or maybe needs some ...
7
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1answer
6k views

Proving that product of two Cauchy sequences is Cauchy

Given that $x_n$ and $y_n$ are Cauchy sequences in $\mathbb{R} $, prove that $x_n y_n$ is Cauchy without the use of the Cauchy theorem stating that Cauchy $\Rightarrow$ convergence. Attempt: Without ...
2
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3answers
957 views

Prove that $a_n=1+\frac{1}{1!} + \frac{1}{2!} +…+ \frac{1}{n!}$ converges using the Cauchy criterion

Any tips on how to approach these kind of proof problems when a factorial is included? Here is what I've tried, By the Cauchy criterion the sequence converges if for every $\varepsilon>0$ there ...
2
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5answers
640 views

Prove the following sequence is a Cauchy Sequence

Let $\{X_n\}$ be the sequence defined recursively by $x_1=2$ and $x_{(n+1)}=(x_n/2)+(5/x_n)$. Prove that $\{x_n\}$ converges and find the limit of the sequence. I understand the definition of a ...
1
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1answer
192 views

lemma: Cauchy sequences are bounded.

Lemma: Cauchy sequences are bounded. Proof: Given Cauchy sequence $(s_n)$, for $\varepsilon=1$ we obtain $N\in\Bbb N$ such that $m, n > N$ implies $|s_n − s_m| < 1$. (2) implies $|s_n − s_{N+1}...
1
vote
1answer
3k views

The space of continuous functions $C([0,1])$ is not complete in the $L^2$ norm

I am trying to prove that the Euclidean Norm/inner product on $C([0,1])$ does not give rise to a complete metric space. To do this I am trying to find a Cauchy Sequence which does not converge in $C([...
1
vote
0answers
34 views

Prove that the system of real numbers is complete by Cauchy sequence

Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! Let $\mathcal{C}$ be the set of Cauchy sequences of rationals. We define an ...
0
votes
1answer
573 views

Proof that the series expansion for exp(1) is a Cauchy sequence

Consider the series expansion for the exponential function at x = 1: $$a_n := \sum\limits_{i = 0}^n{\frac{1}{i!}}$$ I want to prove that this is a Cauchy sequence, using the remainder formula for ...
6
votes
1answer
215 views

Proving a sequence is Cauchy given some qualities about the sequence

I've got a sequence $x_n$ such that I've proved $b\leq x_n \leq c$, and $|x_{n+1}-x_{n}|\leq \frac{4}{9}|x_n-x_{n-1}|$ However I'm not very familiar with Cauchy sequences, so I don't know how to ...
3
votes
2answers
1k views

Showing that a totally bounded set is relatively compact (closure is compact)

I have been tasked with showing that for a metric space $(X,d)$, a subset $E \subseteq X$ is relatively compact $\iff$ $E$ is totally bounded. I believe I have shown the forward implication $(\...