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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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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?
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3answers
2k views

Why do we want complete spaces? We don't we just use closed spaces?

Why do we care about the notion of a space being complete? Why don't just consider closed spaces? If the space is closed we know that the limits of a sequence exist and are in the set which is a ...
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2answers
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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 ...
24
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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: ...
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 ...
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 ...
18
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5answers
20k views

What is the difference between Cauchy and convergent sequence?

I am really confused. I will appreciate if somebody can help me to define the difference between Cauchy and convergent sequence. Many thanks.
18
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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)<\...
18
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3answers
500 views

Recurrence relations and limits, tough.

I would like a hint for the following, more specifically, what strategy or approach should I take to prove the following? Problem: Let $P \geq 2$ be an integer. Define the recurrence $$p_n = p_{n-1} +...
17
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4answers
1k views

Why does $ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$ converge to an irrational number?

There is a problem in my textbook that goes like this $$ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$ and $$a_0 =1$$ for all $n\ge1$. It is monotonically decreasing sequence of rational ...
17
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2answers
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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 ...
15
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3answers
992 views

Counterexample: Continuous, but not uniformly continuous functions do not preserve Cauchy Sequences

I want to prove this: There exists a continuous function $f:\mathbb{Q}\to\mathbb{Q}$, but not uniformly continuous, and a Cauchy sequence $\{x_n\}_{n\in\mathbb{N}}$ of rational numbers such that $\{f(...
14
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5answers
988 views

Fastest way showing the limit exists without finding the limit?

Let $\{a_n\}$ and $\{b_n\}$ be two integer sequences such that $a_1=b_1=1,$ \begin{align*} a_n=a_{n-1}+b_{n-1},\qquad\qquad b_n =r\,a_{n-1}+b_{n-1}. \end{align*} Where $r$ is a natural number $>1$...
14
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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 >...
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5answers
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How is the sequence 1, 1.4, 1.41, 1.414 generated?

In many of the standard textbooks discussing Real Numbers, the Cauchy sequence that converges to $\sqrt{2}$ is given as 1, 1.4, 1.41, 1.414, 1.4142, ... or 2, 1.5, 1.42, 1.415, 1.4143, ... My ...
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1answer
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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 ...
11
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4answers
2k views

Can a Cauchy sequence converge for one metric while not converging for another?

Is there an easy example of one and the same space $X$ with two different metrics $d$ and $e$ such that one and the same sequence $\{x_n\}$ is a Cauchy sequence for both metrics, but converges only ...
10
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7answers
2k views

How can a Cauchy sequence converge to an irrational number?

I am a physics major and would like to clear a confusion regarding complete metric spaces. I am quoting the definition of a Cauchy sequence from wikipedia below Formally, given a metric space $(X, ...
10
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4answers
931 views

Definition of Cauchy Sequence

I have a question regarding the definition of a Cauchy sequence of a sequence in a metric space. The definition I learned and that is consistent with Wikipedia defines a sequence $(x_n)_{n=1}^\infty$ ...
10
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3answers
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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 ...
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2answers
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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 $f: (a, b) \rightarrow \mathbb{R}$. ...
10
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2answers
139 views

What sequences are Cauchy in all metrics for a given topology?

Different metrics for the same topology can have different sets of Cauchy sequences. But I'm interested in what sequences are Cauchy in every metric for a given topology. For a completely metrizable ...
10
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2answers
666 views

Do uniformly continuous functions map complete sets to complete sets?

Let $f: (M, d) \rightarrow (N, \rho)$ be uniformly continuous. Prove or disprove that if M is complete, then $f(M)$ is complete. If I am asking a previously posted question, please accept my ...
9
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1answer
685 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 ...
9
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2answers
790 views

Proving completeness of Nikodym Metric

I'm trying to prove completeness directly of the metric given by $d(A, B) = \mu (A \triangle B)$ on a finite measure space $(X, M, \mu)$. Edit: I should make clear that I'm referring to completeness ...
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}, ...
8
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2answers
377 views

Quasi-Cauchy sequences

Sequence $(x_n)$ is called quasi-Cauchy if $\lim_{n\rightarrow\infty}|x_{n+1}-x_n|=0.$ I need help proving the following theorems: Quasi-Cauchy sequence of real numbers is Cauchy if and only if it ...
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3answers
3k views

Prove that the distance between 2 Cauchy sequences is convergent.

Here is the exact question: Let $(S,d)$ be a metric space. Let $(p_n)$ and $(q_n)$ be two Cauchy sequences in $(S,d)$(note that these two sequences are not necessarily convergent since $(S,d)$ is ...
8
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1answer
288 views

How to find closed form formula for a sum

I am a PhD student in electrical engineering. I need to find a closed form formula for the following series: $$\sum_{k=1}^{\infty}\frac{1}{2}A_k^2e^{-k^2\sigma_m^2}(e^{k^2\sigma_m^2}-1)$$where $A_k= \...
7
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5answers
620 views

Is every converging sequence the sum of a constant sequence and a null sequence?

Let $a_n$ be any sequence converging to $a$ when $n \to \infty$. Can you rewrite $a_n$ so that it is the sum of two other sequences? $$a_n=b_n + c_n,$$ with $b_n=b$ for every $n \in \mathbb{N}$ and $...
7
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1answer
3k views

Every bounded sequence is Cauchy?

I know that every Cauchy sequence is bounded, but is the reverse true?
7
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3answers
19k views

How to generate a Cauchy random variable

How do I calculate a Cauchy random variable and how do I calculate the probability mass function to show it is "heavy tailed"
7
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3answers
1k views

Is it possible to prove that all Cauchy sequences of real numbers converge without using the Bolzano-Weierstrass theorem?

Question: Prove that a sequence of real numbers is convergent if and only if it is a Cauchy sequence. I'm currently learning real analysis through an inquiry based course, and I'm trying to prove the ...
7
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1answer
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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. ...
7
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1answer
138 views

Show that $S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}$ converges for $n\in\Bbb N,\ x \in\Bbb R$ without using Taylor series.

Given a sequence $\{S_n\}$, $n\in\Bbb N$: $$ S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!} $$ Prove that $S_n$ converges for all $x\in\Bbb R$. Please note that i know $S_n$ is a simple ...
7
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2answers
668 views

How to prove that the both definition of completeness of $\mathbb{R}$ are equivalent?

In the definition of completeness of a set, in particular $\mathbb{R}$, I have seen the following definitions: Dedekind: Every non-empty bounded of subset has a least upper bound (with respect ...
7
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2answers
3k views

Cauchy convergence in probability implies the existence of a (finite a.e.) limit $X$

Cauchy convergence of a sequence $X_n$ of random variables in probability implies the existence of an $X$ (finite a.e.), such that $X_n$ converges to $X$ in probability. The problem's hint suggests ...
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 ...
7
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2answers
286 views

Prove a sequence is a Cauchy and thus convergent

Suppose that $0<\alpha<1$ and that $\{ x_n\}$ is a sequence which satisfies $$|x_{n+1}-x_n| \le \alpha^n$$ $$n= 1,2,....$$ Prove that $\{x_n\}$ is a Cauchy sequence and thus converges. ...
7
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1answer
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Given $\Sigma a_n$ diverges show that $\Sigma \frac{a_n}{1+a_n}$ diverges. [duplicate]

Intuitively speaking, I first thought that if the series $\Sigma a_n$ is divergent then $$\lim_{n \to \infty} a_n \ne 0$$ therefore it was clear that $\Sigma \frac{a_n}{1+a_n} $ would be divergent, ...
7
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1answer
98 views

A peculiar characterization of open balls in a Banach space

Let $E$ be a Banach space and $U$ be a bounded open subset of $E$. Suppose that for any $x,y\in U$, there exists some open ball $B$ such that $\{x,y\}\subset B\subset U$. Prove that $U$ is ...
7
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1answer
492 views

Banach space with respect to two norms must be Banach wrt the sum of the norms?

Let $X $ be an infinite dimensional $R$-vector space, suppose that $||\cdot||_1$ and $||\cdot||_2$ are two norms that makes $X$ into a Banach space. Let $||\cdot||_3 = ||\cdot||_1 + ||\cdot||_2 $ ,...
7
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1answer
122 views

The limit points of $\{\sin (n+n^2)\mid n\in\mathbb N\}$

We can prove that $\exists \{c_i\}_{i=1}^{\infty} \subseteq \mathbb N$ and $c_1<c_2<c_3<...$ such that $\lim_{i\rightarrow \infty}\sin (c_i^{\alpha})=c$, $\forall c\in [-1,1]$,where $\alpha $...
7
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1answer
227 views

Find the value of the infinite product: $\prod\limits_{n=1}^{\infty}\left(1+\frac{1}{3^n}\right)$

I am Anay, here is a problem I am stuck with: $$x = \prod\limits_{n=1}^{\infty }\left ( 1 + \frac{1}{3^n} \right )$$ The task is to find the value of $x$. (obviously, we aren't supposed to have ...
7
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1answer
189 views

Is this sequence necessarily Cauchy?

Let $(X,d)$ be a metric space and $(x_n)$ a sequence in it such that for all $i,j,n\in\mathbb{N}_{>0}:$ $$d(x_i,x_j)<n+1\Rightarrow d(x_{i+1},x_{j+1})<n$$ $$d(x_i,x_j)<\frac{1}{n}\...
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 ...
6
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6answers
1k views

Making sure if it is Cauchy

In my real analysis exam I had a problem in which I proved that $|x_{n+1} - x_n|\lt {a^n}$ for all natural numbers $n$ and for all positive number $a\lt 1$ then $(x_n)$ is a Cauchy sequence. This was ...
6
votes
2answers
762 views

Is every Cauchy sequence in a non-complete metric space convergent?

A metric space $X$ is called complete if every Cauchy sequence in $X$ has a limit in $X$. For a non-complete metric space $X$, can we say that every Cauchy sequence is convergent? (even though the ...
6
votes
4answers
516 views

Cauchy sequences - can we control the rate at which elements “get closer”?

In Simon & Reed's book Methods of Modern Mathematical Physics, it is proven in chapter 1 (Theorem 1.12) that $L^1$ is complete (Riesz-Fisher theorem). The proof starts off as follows: Let $f_n$ ...
6
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1answer
606 views

challenge problem: show this sequence is convergent.

had this difficult question from a textbook, and I haven't been able to figure out the solution. say we have a sequence of bounded real numbers $a_n$ such that $2a_n \leq a_{n-1} + a_{n+1} \forall n\...