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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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How can we construct reals from Cauchy sequences if the definition of Cauchy makes uses reals?

My understanding the definition of a Cauchy sequence is that it is a sequence $a_n$. Which has the property. $$\forall \epsilon >0 \exists N \in \mathbb{N} : \forall n,m \geq N | a_n - a_m |< \...
Q the Platypus's user avatar
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Sign permanence of locally Lipschitz functions calculated on a sequence

Suppose I have a sequence $a_k(m)>0$ with $m\in\mathbb{N}$ such that, given $k\in\mathbb{N}$ and $p\geq 1$, I can show that $$|a_{k+p}(m)-a_k(m) |\leq \frac{m^2}{k^2}$$ with $a_{k+p}<a_{k}$. I ...
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Recursive Sequence convergence

Let $(b_n)_n$ be a nonnegative sequence and such that $$\lim_{n\rightarrow\infty}b_{n+1}^2-b_n=a>0$$ Show that $(b_n)_n$ is convergent My approach is write $a_n=b^2_{n+1}-b_{n}$ and there exists an ...
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Cauchy product of trapezoidal sequences is also trapezoidal

Let $(a_0,a_1,\dots,a_n)$ be a sequence of positive integers such that $a_i=a_{n-i}$ for all $1\leq i\leq n$, and let $m=\lfloor \frac{n}{2}\rfloor$. We say the sequence satisfies the trapezoidal ...
Chard's user avatar
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2 answers
102 views

Imagining Rational Cauchy Sequences as Dancing Around a Real Number Instead of Converging to One

I'm trying to build my intuition regarding the Cauchy-sequence construction of the reals. Essentially, do you think that it is more accurate to visualize a real number as being defined by sequences of ...
mouldyfart's user avatar
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1 answer
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$\sum_1^{+\infty} a_j^2 < \infty$ if and only if $\psi_n$ is a Cauchy sequence?

Let $(\xi_j)_j$ be a complete orthonormal basis of $L^2(\mathbb{R})$, I want to prove , Given that $\sum_1^{+\infty} a_j^2 < \infty$, the sequence $$\psi_n(x) = e^{-\frac{1}{2}\sum_{j=1}^n a_j\...
Houssem Ajili's user avatar
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Adjacent vs. All-Possible Distances in the Definition of a Cauchy Sequence

I'm currently learning about Cauchy sequences, and I'm trying to build my intuition regarding its definition. My question is on how we can motivate why we consider the distances between all points and ...
mouldyfart's user avatar
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3 answers
58 views

if $n\geq N\implies |x_{n+1} - x_n| < \epsilon$ then $x_n$ converges

Let $x_n$ be a sequence of real numbers in the interval $[a,b]$ for some $a,b \in \mathbb R$. If $x_n$ has the property: For all $\epsilon > 0$, there is $N \in \mathbb N$ such that for all $n\geq ...
Eduardo Magalhães's user avatar
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Prove that the sequence $(A_n(f))$ is convergent in the normed space $C[0,1]$. Prove that the sequence $(A_n)$ is not convergent in the operator norm.

Let $C[0,1]$ be a normed space equipped with the norm $\|\cdot\|_\infty$, and let for every $n \in \mathbb{N}$, the mapping $A_n$ be given by the prescription $ (A_n(f))(x) = \begin{cases} f(x), &...
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Tao: Positive/negative real numbers are defined as positively/negatively bounded away Cauchy sequences. Those who start negative and turn positive -?

Quote from Tao's Analysis 1: Definition 5.4.3 A real number $x$ is said to be positive if and only if it can be written as $x = \lim_{n \to \infty} a_n$ for some Cauchy sequence $(a_n)_{n=1}^{\...
Community_Digest's user avatar
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Show that $(\operatorname{Im} A)^\perp = 0$ and from this deduce that $A$ is a homeomorphism.

Let $H$ be a Hilbert space and $A \in \mathcal{B}(H)$ be a self-adjoint operator such that $|x| \leq |Ax|$ for every $x \in H$. Show that $(\operatorname{Im} A)^\perp = 0$ and from this deduce that $A$...
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2 answers
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Rationals are incomplete and naturals are complete

Why are rationals incomplete and natural complete? I have been going through Analysis textbooks but I don't totally get the reason why. So, naturals are complete because you can divide them into two ...
pdaranda661's user avatar
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I want to prove the following: If $(X, \|\cdot\|)$ is a Banach space, then every Cauchy sequence in $(X, \|\cdot\|_1)$ converges in $(X, \|\cdot\|)$

Let $(X, \|\cdot\|)$ be a normed space, and let $A: X \to X$ be a linear operator. First, I proved that the prescription $\|.\|_1 = \|x - Ax\| + \|Ax\|$ defines a norm on $X$. I want to prove the ...
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$\|p\| = \max_{n \in \mathbb{N}} |a_n p^{(n)}(0)|$, a norm is defined on $X$, where $p^{(n)}$ is the nth derivative of the polynomial $p$

Let $X$ be the vector space of all polynomials with real coefficients and let $(a_0, a_1, \ldots)$ be a sequence of positive real numbers. I was able to show that with the prescription $\|p\| = \max_{...
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2 answers
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Let $X$ be the vector space of all real sequences that have at most finitely many non-zero terms. Is $(X, \| \cdot \|)$ a Banach space?

Let $X$ be the vector space of all real sequences that have at most finitely many non-zero terms. I was able to show that the prescription $\| \{x_n\}_{n \in \mathbb{N}} \| = \max_{n \in \mathbb{N}} |...
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1 answer
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If a random variable converges to zero in probability what can we say about its almost sure boundedness?

First let me start with definitions that I will be using in the question. A sequence of random variables $X_n(\omega)$ converges to zero in probability if for any $\epsilon>0$, and any $\delta>...
curiosity's user avatar
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Prove that $(H,${$.,.$}$)$ is a Hilbert space.

Let $U \in B(H)$ be a unitary operator and $A \in B(H)$ be a self-adjoint operator on a Hilbert space $(H,\langle ., . \rangle)$. Introduce the mapping {$.,.$} $:H \times H \rightarrow H$ defined by {$...
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2 answers
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Let f monotone function in $\mathbb{R}$. Then $x_n$ Cauchy implies $f(x_n)$ Cauchy sequence [closed]

I have to prove is true the following, given $f:\mathbb{R}\to\mathbb{R}$, f monotone in $\mathbb{R}$: $\forall x_n$ sequence in $\mathbb{R}$, if $(x_n)$ is a Cauchy sequence, then $f(x_n)$ is a Cauchy ...
axi's user avatar
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If $f$ is differentiable at $a$, does it imply that $\lim\limits_{x,y\to a\atop x\ne y} \frac{f(x) - f(y)}{x-y} =f^{\prime}(a)$? [duplicate]

Let $a \in \mathbb{R}$, $I \subseteq \mathbb{R}$ be a neighborhood of $a$, $f: I \rightarrow \mathbb{R}$ a function which is differentiable at $a$. Want (either/or) : A function $f$ for which there ...
Colver's user avatar
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4 answers
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How to prove the sequence of function $\{f_n\}$ is Cauchy but not convergent? (Details Below)

Problem Consider the space $C[-1,1]$, together with the norm defined by $\|f\|_1 = \int_{-1}^1|f|d\lambda$ (where $\lambda$ is the Lebesgue measure). For each $n$ define a function $f_n:[-1,1]\to\...
Beerus's user avatar
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2 votes
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Cauchy but not convergent sequence on $C[-1,1]$

Consider $C[-1,1] = \{ f: [-1,1] \rightarrow \Bbb{R}: f$ is continuous $\}$, the inner product $\langle f,g \rangle = \int_{-1}^1 f(x)g(x) \, dx$ and $\lVert \cdot \rVert$ the induced norm. Then, the ...
Daniel García's user avatar
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1 answer
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Unable to come up with correct bounds for showing sequence convergence. Analysis I, Terence Tao, Theorem 6.1.19, part (g)

I was trying to prove the following (part (g) of the Theorem): Theorem 6.1.19 (Limit Laws). Let $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ be convergent sequences of real numbers, and let $x,y$...
Paul Ash's user avatar
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Need help with proof of an auxiliary result in Analysis I, Terence Tao

I attempt to prove the below result which is needed to prove an overall result in Analysis I, Terrance Tao. For reference, that overall result is Theorem 6.1.19 (Limit Laws) part (e). Any sequence ...
Paul Ash's user avatar
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1 vote
1 answer
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Does continuity on a closed set preserves Cauchy sequences?

Let $f$ be a real-valued continuous function on $D=[0,\infty)$. Does $f$ preserve Cauchy sequences, i.e., if $\{x_n\}$ is Cauchy, is $\{f(x_n)\}$ Cauchy? Since $D$ is closed, the Cauchy sequence has a ...
user488240's user avatar
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Prove that $(X, d)$ is a complete metric space where $X$ is the set of all real sequences and $d: X \times X \to \mathbb{R}$ defined by...

I am given a metric space $(X, d)$ where $X$ is the set of all real sequences and $d: X \times X \to \mathbb{R}$ the metric defined by $$ d(x, y) = \begin{cases}(\sup\{n \in \mathbb{N}: x_k = y_k \...
Felix Gervasi's user avatar
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1 answer
21 views

Complete space relative to specific norm

Let $X=\{f\in C^1([0,\infty, \mathbb{R})): \lim_{t\to\infty}\frac{f(t)}{1+t}=\lim_{t\to\infty} f'(t)=0\}$. How to prove that $X$ is Banach space, if norm is defined as $\|f\|=max(\sup_{t\geq 0}\frac{|...
alans's user avatar
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1 answer
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Completeness preserved under specific homeomorphism

Problem: Let $(X,d)$ be a complete metric space, $(Y,d')$ a metric space, and $f\colon X\to Y$ a homeomorphism, such that there exists a $c>0$, for which $$c\cdot d(x,y)\leq d'(f(x), f(y)).$$ Show ...
categoricallystupid's user avatar
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Baby Rudin Chapter 3 Exercise 14d

I'm working through baby rudin chapter 3 exercises, and I was just curious to see if my proof also works for exercise 14d. Rudin gives: $\sigma_n=\frac{s_0+s_1+...+s_n}{n+1}, (n=0,1,2...)$, assume $\{\...
rudinable's user avatar
2 votes
1 answer
49 views

Understanding a proof that a uniformly Cauchy sequence of continuous functions $\{f_n\}_n$ converges uniformly to a limit function $f$

Suppose $\{f_n\}_n$ is a uniformly Cauchy sequence of continuous function $f_n:\mathbb{R}\to S$ for a complete metric space $S$. I am trying to understand the "standard" proof that the ...
Cartesian Bear's user avatar
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1 answer
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Proving a sequence of functions is Cauchy, but not convergent in a normed space.

$$f_n : [0,2] \to \Bbb R : x \mapsto f_n(x)= \begin{cases} x^n & \text{if } x \in[0,1] \\ 1 & \text{if } x \in (1,2] \end{cases}$$ for $$n \in \Bbb N$$. Show that the sequence $$f_n$$ is a ...
clumsyclot's user avatar
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1 answer
35 views

Convergence in measure and almost uniform Cauchy convergence almost everywhere

I,m trying to prove the following: Let $(X, \Omega, \mu)$ be a measure space and $f,f_n: \Omega \to \mathbb{K}$, $n \in \mathbb{N}$ be measurable functions where $\mathbb{K} = \mathbb{R}$ or $\mathbb{...
Kham Bodrogi's user avatar
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0 answers
32 views

Show that the sequence $(f_n)_{n \in \mathbb{N}} \in (C^0([0,2]),||\cdot||_1)$, is a Cauchy-sequence, but does not converge

We consider the sequence $(f_n)_{n \in \mathbb{N}}$ defined by $f_n : [0, 2] \rightarrow \mathbb{R}$ as follows: $f_n(x) =\begin{cases} x^n, & \text{if } 0 \leq x \leq 1 \\ 1, & \...
j.primus's user avatar
3 votes
1 answer
90 views

Proposition 5.4.9. Analysis I - Terence Tao.

Proposition 5.4.9 (The non-negative reals are closed). Let $a_1, a_2, a_3, \ldots $ be a Cauchy sequence of non-negative rational numbers. Then $\text{LIM}_{n \to \infty}a_n$ is a non-negative real ...
Paul Ash's user avatar
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60 views

How is it possible that Cauchy sequence represents two different numbers?

In the book on Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, p.4) there is such term as "standard infinitesimals". This ...
Mike_bb's user avatar
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1 vote
0 answers
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How to correctly choose number for Cauchy sequence for infinitesimal functions? Will it be $x=y=[sin(t)]$ or $x=y=[t]$?

I read about Infinitesimal Differential Geometry (http://www.iam.fmph.uniba.sk/amuc/_vol-73/_no_2/_giordano/giordano.pdf, page 4) and there was case with Cauchy sequence for infinitesimal functions: ...
Mike_bb's user avatar
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1 vote
1 answer
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$A_n-A_{n-1} = q^nA_n-aq^{n-1}A_{n-1}$ for Cauchy sequence

For $|q| < 1$, $|t|<1$, then $$1+\sum_{n=1}^{\infty} \frac{(1-a)(\cdots)(1-aq^{n-1})t^n}{(1-q)(\cdots)(1-q^n)}=\prod_{n=0}^{\infty}\frac{(1-atq^n)}{(1-tq^n)}=\sum_{n=0}^{\infty}A_nt^n$$ Fromwhat ...
Nishi's user avatar
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1 answer
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$\mathbb{Q}$ is dense in $K$ if each element in $K$ is the limit of a Cauchy sequence in $\mathbb{Q}$.

Let $K$ be an ordered field containing the field of rationals $\mathbb{Q}$. Prove that $\mathbb{Q}$ is dense in $K$ if and only if every element of $K$ is the limit of a Cauchy sequence in $\mathbb{Q}$...
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2 answers
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Theorem 3.11 (c): Rudin's PMA

I wanted to ask some clarification on one of the proofs in Rudin's PMA, specifically Theorem 3.11 (c). It states that In $\mathbb R^k$, every Cauchy sequence converges. The proof for it is as ...
random math acc's user avatar
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2 answers
172 views

A question on a step in proving $L^\infty$ is complete.

Let $(X,\mathcal{A},\mu)$ be $\sigma$-finite measure space. Claim: $L^\infty$ is complete. Ideas of the proof: (i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\...
William Chuang's user avatar
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0 answers
44 views

Can someone verify if my proof is accurate?

I am only allowed to use the fact that a certain sequence {${x_i}$} that is not equivalent to the zero sequence is cauchy to prove that the reciprocal {$\frac{1}{x_i}$} is also cauchy. i.e. we know ...
Vector's user avatar
  • 377
1 vote
1 answer
49 views

Cauchy sequences over $\mathbb{Q}$: A basic question

Define an equivalence relation on Cauchy sequences in $\mathbb{Q}$ by $(x_n)\equiv (y_n)$ if $\lim (x_n-y_n)=0$. The question I am trying to prove is that If $(x_n)$ and $(y_n)$ are Cauchy sequences ...
Maths Rahul's user avatar
  • 3,065
4 votes
1 answer
85 views

Using only the universal property, prove that $X$ is dense in its Cauchy completion

Long ago, I learned about the Cauchy completion of metric spaces via the usual explicit construction of quotienting the set of Cauchy sequences. For a metric space $X$, let $\hat X$ denote this Cauchy ...
Atom's user avatar
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1 vote
1 answer
50 views

Proving a sequence is not Cauchy

I want to prove that a sequence is not Cauchy by proving something like $$ \exists \epsilon >0 \space s.t \space \space \forall n_0 \in N \space \space \exists n, n+p > n_0 \space=> |x_{n+p} ...
Mmm's user avatar
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2 votes
1 answer
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Given a sequence $a_n$ s.t. forall $\varepsilon>0$ exists $m,n_{0} \in \mathbb{N}$ s.t. $\forall n \geq n_0$ $|a_n-a_m|<\varepsilon$

So I have that question: Suppose a sequence $a_n$ satisfies that forall $\varepsilon>0$ exists $m,n_{0} \in \mathbb{N}$ s.t. $\forall n \geq n_0$ $|a_n-a_m|<\varepsilon$ determine if that ...
oneneedsanswers's user avatar
0 votes
1 answer
93 views

Not decreasing sequence that converges to 0: sufficient condition for the sequence of indexes for which the sequence decreases to be bounded.

Consider a strictly positive sequence $\{u_n\}$ of real numbers (i.e. $\forall n, u_n >0$). Suppose that $\displaystyle \lim_{n \to \infty} u_n = 0$ and that $\{u_n\}$ is not decreasing. Consider ...
Oussama Zekri's user avatar
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1 answer
84 views

Proof to show Sequence is Cauchy correct?

Given the two facts The sequence $\{ x_n \}$ is Cauchy in the $L^2(D)$ norm. For each $x$ we can apply the following inequality $\|x\|_{E(D)} \leq k \|x\|_{L^2(D)}$ where $k > 0$ I want to show ...
k12345's user avatar
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-2 votes
1 answer
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an example that the limit of a converging sequence is not unique [closed]

The following is an example that the limit of a converging sequence is not unique. There are uncountably many numbers in Cantor Set. Thus, there are uncountably many numbers in the segments in Cantor ...
user1286486's user avatar
0 votes
1 answer
46 views

Trying to construct a subsequence

Let $(x_n)$ be a real valued sequence. I want to construct a subsequence $(y_n) $ of $(x_n) $ such that no two consecutive terms of the sequence $( y_n) $ are same and also if $(y_n) $ is Cauchy then $...
Math Lover's user avatar
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How is contradiction with definition of product of two sequences possible?

I read about Infinitesimal Differential Geometry in this source, page 238-239 and I considered one example while I was reading. I was confused because in this example we have contradiction with ...
Mike_bb's user avatar
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2 votes
2 answers
169 views

Asymptotic Analysis of a Sequence Problem: Convergence of $\log(n)a_n$ as $n\to+\infty$

Analyzing the Convergence of $\log(n)a_n$ in a Sequence Problem Consider two positive sequences $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ satisfying the following conditions: $(a_n)_{n \geq 0}$ is a ...
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