Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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31 views

Homeomorphism between two complete spaces.

Let $X,Y$ be complete metric spaces and let $h: X \to Y$ be homeomorphism. If $(x_n)$ is a Cauchy sequence in $X$, is it true that $(h(x_n))$ is Cauchy in $Y$? Instinctively, I would go for no, ...
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33 views

Proving Cauchy sequence and its convergence [duplicate]

As I am having trouble in concluding the question, please look at this also: For a sequence $$1 - \frac{1}{2} +\frac{1}{3}+\cdots +\frac{(−1)^{n−1}}{n} \\ \text{Then}\\ |s_n−s_m| = \left|\frac{(−1)^{m−...
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1answer
17 views

Use of Cauchy general principle of convergence [duplicate]

Question: Prove the convergence of sequence using Cauchy general principle of convergence. $$\frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + . . + \frac{1}{n!}.$$ My attempt: $$ | s_2 - s_1 | = \frac{1}{2!...
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32 views

Determine whether of the following sequences is cauchy and find the limits [closed]

Determine whether of the following sequences is cauchy and find the limits: $x_{1}$=1, $x_{n+1}$=$x_{n}$+1/n+1.
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Is the function space $X=\{u \in H^1(\Omega) : \text{$u$ is continuous at $0$}\}$ complete?

This question is somewhat similar to Form functions that are continuous at one point in L^\infty a Banach space. where the continuity at zero was added to $L^\infty(\Omega)$. This space was indeed ...
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32 views

Why a Cauchy sequence in $W^{1, p}$ is also a Cauchy sequence in $L^p$

$W^{1,p}(I)= \{ u\in L^{p}(I) | \exists g \in L^p(I) :\int_{I}^{}u\varphi'= - \int_{I}^{}g \varphi \}$, where I = (a, b) $\subset \mathbb{R}$ with the norm $||u||_{W^{1,p}}=||u||_{L^p}+||u'||_{L^p}$ ...
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9 views

Prove a sequence converges to x given a condition [duplicate]

I have this question in an analysis book, chapter related to Cauchy sequences. Let ${x_n}$ be a sequence. Suppose that there is an $a \in (0,1)$ such that $$|x_{n+1}-x_n| \leq a^n$$ for all $n \in \...
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37 views

Cauchy sequences under derivative of a differentiable function [duplicate]

Suppose $f : \Bbb R → \Bbb R$ is differentiable on $\Bbb R$, and let $(x_n)$ be a Cauchy sequence. Is the sequence $(f′(x_n))$ also Cauchy? My first thought was to take $f(x)=\log x$, and $(x_n)=1/n$. ...
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13 views

Functional analysis about sequence [duplicate]

Good evening. Yes, I know my title is very unimaginative, but I was not able to find a way to summarize my question. I have a school assignment that must be written as an article, and my topic is ...
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25 views

Proof that the space of vector valued function is complete

Let $\Omega \subseteq \mathbb{R}^{n}$ be open and $(\mathbb{E},\|\|)$ a Banach space and let $C^{\infty}(\Omega, \mathbb{E})$ denote the space of all smooth functions from $\Omega$ to $\mathbb{E}$. ...
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48 views

Prove that seqences {cos(n)}, {sin(n)} diverge.

I made a proof using isometry, but I'm not sure. Here is the proof: Let A be a 2×2 matrix whose first row is equal to (cos(1), -sin(1)) and second row is equal to (sin(1), cos(1)). And define the ...
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70 views

Convergence of series with logarithm upon polynomial as terms [duplicate]

Consider the following sequence $a_n=\dfrac{\ln n}{n^p}$ where $p>1$. I want to check for the convergence of the series of that sequence i.e. $\displaystyle \sum_{n=1}^{\infty}a_n$. I intuitively ...
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19 views

Quasi-increasing sequences proof-verification

I am self-studying the book Understanding Analysis by S. Abbott and I tried to solve the exercise 2.6.6 (c) without relying on Bolzano-Weierstrass Theorem as they did here. Such exercise requires the ...
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1answer
50 views

Prove that if a sequence is Cauchy, then it is bounded.

I have a question about the proof of Cauchy implying boundness. The proof argues that after $N$ terms, the sequence is bounded, i.e. we could have $|s_n - s_N| < 1, \forall n > N \Rightarrow s_n ...
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1answer
62 views

convergence of the series $\sum_{n=1}^{\infty}(1/n^{\alpha})(\int_{0}^{\frac{\pi}{4}}\tan^{n}tdt)x^{n}$

We are given $$\displaystyle W_n =\int_{0}^{\frac{\pi}{4}}\tan^{n}tdt.$$ Find the radius of convergence of the power series $$\displaystyle\sum_{n=1}^{\infty}\frac{W_n}{n^{\alpha}}x^n$$ in terms of ...
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1answer
60 views

Example of a sequence on an infinite-dimensional vector space with respect to different norms [closed]

I am stuck with the following problem: Give example of an infinite dimensional vector space V and two norms $\theta$ and $\rho$ on V and the sequence $\{x_{n}\}_{n\geq 1}$ of V such that: The ...
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1answer
51 views

For which $p \in [1, \infty]$ the sequence is Cauchy

Let $f_n (x) = \frac{n}{1+n\sqrt{x}}$, for $x \in (0,1)$. I am asked to find for which $p \in [1, \infty]$ $(f_n)_n$ is a Cauchy sequence with respect to $L^p$ norm. First of all, I observed that $(L^...
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21 views

Is $u_n:= \sin(\sqrt{n})$ an example of a bounded, non-convergent real sequence so that the difference between the successive terms go to zero? [duplicate]

I'm looking for an example of a bounded sequence $(u_n),$ so that $|u_n-u_{n-1}|\to 0,$ but $(u_n)$ is not convergent. For unbounded such sequence, it's easy to construct such an example: $$u_n:= \...
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1answer
26 views

Prove the succession $\{x_n\} = \{q^n,\ |q|<1\}$ is a Cauchy succession

So far I've only proven this fact given both $\{x_n\},\ \{x_m\}$ are either positive or negative: Let $\varepsilon >0$. I shall prove that given $\varepsilon = 2|q|^N$, the succession is bounded by ...
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0answers
22 views

Searching for a sequence that shows something isn't a complete metric space [duplicate]

Hey I have to find a Cauchy-Sequence that converges in $(\mathbb{R} ,d_{|\cdot |})$ (absolute value as metric) and is still a Cauchy- Sequence with regards to a metric defined as $d(x,y):=|\frac{x}{1+|...
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1answer
111 views

Prove that $\Gamma=\{\gamma:K\longrightarrow (Y, \| . \|) : \gamma\; \text{is continuous and} \; \gamma|_{K_0}=\gamma_0\}$ is a complete metric space.

QUESTION: Let $(X, \|.\|)_X$ be a Banach space and $Y\subset X$ a subspace, which is itself a Banach space endowed with a norm $\|.\|_Y$ such that $\|y\|_X\leq \|x\|_Y$ for every $y\in Y$. Let $K$ be ...
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1answer
56 views

Proving that the sequence $x_n = \frac{n+1}{3n-2}$ is Cauchy.

The definitions I've been given for Cauchy sequence are 1)$\forall \epsilon>0 ,\, \exists N \in\Bbb{N} \,$ $\ni $ $\forall n,m>N \ $ , it follows that $|x_m-x_n|<\epsilon$ 2)$\forall \...
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1answer
71 views

Prove that the given sequence converges. [duplicate]

The given sequence is: $ x_n = \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} $ I want to first try showing that it is a cauchy sequence. The definitions I have for cauchy sequence are 1)$\...
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2answers
57 views

Is $\{f(x_n^2)\}$ a Cauchy sequence if $|x_n-x_{n+1}|<\frac{1}{10^n}$ and $f$ is continuous?

Let $\{x_n\}$ be a sequence in $\Bbb R$ that satisfies $$|x_n-x_{n+1}|<\frac{1}{10^n}\tag{$*$}$$ for each $n$, and let $f:\Bbb R\to\Bbb R$ be continuous. I'd like to determine whether $\{f(x_n^2)\}$...
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0answers
37 views

One problem in Proof of Bessel's inequality in Functional Analysis.

Problem Prove Bessel's inequality using Lenma below. Bessel's Inequality Let $H$ be Hilbert space on $\mathbb C$, $(\cdot , \cdot)$ be an inner product in $H$, and $\{ e_k \}_{k\in \mathbb Z}$ be an ...
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2answers
38 views

How to prove that $ \frac{n+1}{4n^2+3}$ is a Cauchy sequence?

By definition, I need to show that for any $\epsilon \gt 0$ there exists $N \in \mathbb{N}$ such that for any $m,n \gt N$: $ \lvert a_n - a_m \rvert \lt \epsilon$ So I write, $ |\frac{n+1}{4n^2+3} - \...
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1answer
37 views

Prove sequence convergence using Cauchy Sequences

How should I prove convergence using Cauchy sequences $$ \sum_{k=1}^{n} \frac{\sin({k^{3}+1)}}{(4k+1)(4k+5)} $$ I tried starting with the definition $\forall\varepsilon>0,\exists N\in\mathbb{N}\ \...
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1answer
60 views

Prove that $(1+\frac{1}{n})^n$ is a Cauchy sequence

I have to prove that ($\mathbb{Q}$,d) is not a complete metric space with a counterexample; so, I have to find a sequence which approacches to an irrational number and must be a Cauchy sequence. I ...
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1answer
56 views

Prove that the metric space is Cauchy: $d(x,y)=|\frac{x}{3+x}-\frac{y}{3+y}|$

I am trying to show that metric space $([0,4],d)$ with $d(x,y)=|\frac{x}{3+x}-\frac{y}{3+y}|$ is complete. Using the theory: A metric space (X,d) or normed space (X,‖∙‖) is called complete if every ...
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1answer
38 views

Sow that if $\mathcal{H}$ is not finite-dimensional, then there exists a sequence in $B$ that does not have a convergent subsequence.

Consider a Hilbert space $\mathcal{H}$ with inner product $\langle \cdot, \cdot \rangle$ and norm $|| \cdot ||$. Consider the set $$B = \{x \in \mathcal{H} | ||x|| \leq 1 \} $$ Then I have to show ...
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1answer
63 views

Proof that sequence is a Cauchy sequence.

Let $s_0$ and $s_1$ be arbitrary real numbers. Suppose $\alpha, \beta >0$ such that $\alpha + \beta = 1$. Define the sequence for $n \geq 1$, $$s_{n+1} = \alpha s_{n} + \beta s_{n-1}. $$ I'm trying ...
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1answer
26 views

Understanding the class of all convergent sequences in a discrete space. [duplicate]

Prove: $X$ with the discrete metric $d$ where, $d(x,y)=\begin{cases} 1,&x\ne y\\0,& x=y\end{cases}$ $ (x_n)$ is convergent if and only if it is constant for a sufficiently large $ n$. I ...
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1answer
57 views

If the difference between two consecutive elements tends to 0, then the sequence converges? Mistake in proof

I tried proving that if $a_{n+1}$ - $a_n$ $\rightarrow$ $0$, then $a_n$ converges. Now I searched for this example here and saw many counterexamples, yet I don't understand where my proof went wrong. ...
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3answers
77 views

Let the sequence $\{a_n\}$ be defined as $a_1=1$ and $a_{n+1} = \frac{6a_n+3}{a_n+4}$. Show that $a_n \lt 3$ and the sequence is increasing.

Let the sequence $\{a_n\}$ be defined as $a_1=1$ and $$a_{n+1} = \frac{6a_n+3}{a_n+4}$$ Then I'm asked to show : $1)$ $a_n \lt 3$. $2)$ Assuming $a_n \lt 3$, show that the sequence is increasing. For ...
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41 views

If $|x_{n+1}-x_n| < |x_n-x_{n-1}|$, then the sequence $\left\{\frac{x_n}{\log(n)}\right\}$ is convergent.

Question - Is the following statement true : If $|x_{n+1}-x_n| < |x_n-x_{n-1}|$ for all $n\geq 2$, then the sequence $\left\{\frac{x_n}{\log(n)}\right\}$ is convergent. I know that if $|x_{n+1}-...
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1answer
22 views

Shortcut to recognize a cauchy sequence?

I know kinda shortcut for uniformly continuous functions (which is Cauchy criterion) by seeing if the derivative of the function is bounded or not, so I was wondering if there is a shortcut or trick ...
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1answer
55 views

Is this sequence Cauchy?

Consider a sequence of functions $f_{n}:K=[0,1]\times [0,1]\to \mathbb{R},\;f_{n}(x,y)=\dfrac{\ln(e^{nx}+e^{ny})}{n}$. Is this sequence Cauchy (in $(C(K),\;\rho)$, where $\rho(f,g) = \sup\limits_{x\in ...
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0answers
45 views

If any two sequences converge to the same limit, prove they are concurrent.

The sequences $\{x_n \},\{y_n \}$ are called concurrent if and only if $d(x_n,y_n) \to 0$ as $n \to \infty$. We denote this property as $\{x_n \} \approx \{y_n \}$. Prove if any two sequences converge ...
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0answers
68 views

$\text { Prove } u_{n}=\frac{\cos 5}{5}+\frac{\cos 5^{2}}{5^{2}}+\cdots+\frac{\cos 5^{n}}{5^{n}} \text { is convergent in } \mathbb{R} \text {. } $

The way the book proved it is this: Its idea was to prove that the sequence is Cauchy then it follows that it is convergent in $\mathbb{R}$ ; But my way that I want to check if it's right or wrong is ...
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1answer
40 views

Question About Equivalence

I am wondering if the following statement is true, and if not, why not? If $l-\epsilon < a_n < l + \epsilon$ for all $\epsilon > 0$, then $a_n = l$. This is in the context of Cauchy Sequences....
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53 views

$L^1_{loc}(\mathbb{R})$ is a Fréchet space

Show that the space $L^1_{\text{loc}}(\mathbb{R})$ endowed with a system of seminorms $$ q_n(f) = \int_{-n}^n |f| \quad f \in L^1_{\text{loc}}(\mathbb{R}), n \in \mathbb{N} $$ is a Fréchet space. We ...
2
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1answer
70 views

If $\sum_n\|x_n-x_{n-1}\|^2<\infty$, is $(x_n)$ Cauchy?

Let $(E,d)$ be a metric space and $(x_n)_{n\in\mathbb N}\subseteq E$. We can easily show that if $(\varepsilon_n)_{n\in\mathbb N}\subseteq[0,\infty)$ is summable and $$d(x_n,x_{n+1})\le\varepsilon_n\;\...
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0answers
113 views

If $(f_n)$ is Cauchy and converges a.e. to $f$, there is a Cauchy subsequence $(f_{\psi (n), x})_n$ converging to $f_x$ in $\mathcal L_1$ and a.e.

In generalizing Fubini's theorem to functions on Banach space, I encounter below result that takes me a great deal of time to prove. The proof is delicate for me because we play with $\mathcal L_1$ as ...
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1answer
68 views

Proof that subspace of $L^2\{\mathbb{R}\}$ is Hilbert space

I would like to understand whether or not the following subspace of $L^2{[0,\infty)}$ is also a Hilbert space. Let $H[0,\infty]$ be a subspace of $L^2{[0,\infty)}$, with inner product $\langle f, g\...
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5answers
135 views

Prove that the sequence $(a_n)$ is Cauchy and find the limit.

Let us define a sequence $(a_n)$ as follows: $$a_1 = 1, a_2 = 2 \text{ and } a_{n} = \frac14 a_{n-2} + \frac34 a_{n-1}$$ Prove that the sequence $(a_n)$ is Cauchy and find the limit. I have proved ...
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0answers
60 views

$\left\{\frac{1}{n}\right\}$ converges to $\frac{1}{2}$ in some metric space $(\mathrm{X}, \mathrm{d})$

Consider the space $X=[0,1]$ Then A) $\left\{\frac{1}{n}\right\}$ converges to $\frac{1}{2}$ in some metric space $(\mathrm{X}, \mathrm{d})$ B) $\left\{\frac{1}{n}\right\}$ converges to 1 in some ...
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0answers
68 views

Proof the sequence converges - contractive sequence, solution check

Suppose we have a recursive sequence $a_n$ with $a_1=1$ and $a_{n+1}=1+1/a_n$. Prove that $a_n$ is convergent and determine the limit of the sequence. So far I have: First $(a_n)>0,\forall n\in\Bbb ...
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0answers
26 views

A cauchy sequence that needs geometric sum to calculate

Let $q\in [0,1)$ and $n \ge 2$ Proof that the sequence $$|x_{n+1}-x_n|\le q|x_n-x_{n-1}|$$ is a cauchy sequence. I had no idea how to start so i looked into the solution sheet, in which they mentioned ...
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0answers
14 views

Finding what to add and subtract in absolute value to prove convergence of seqences

Let's say I have 2 convergent sequences $a_n$ and $b_n$ and I want to prove that then sequences $a_nb_n$, $\frac{a_n}{b_n}$, $a_n+b_n$, $a_n-b_n$, or any other similar provable sequence is convergent. ...
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1answer
72 views

$\{ X_n \}$ is a sequence such that $\vert X_{n+1} - X_n \vert \lt \frac{1}{2^n}$ for all $n$. Then show that $\{ X_n \}$ is Cauchy.

$\{ X_n \}$ is a sequence such that $\vert X_{n+1} - X_n \vert \lt \frac{1}{2^n}$ for all $n$. Then show that $\{ X_n \}$ is Cauchy. I know that I want to show: $\vert X_{m} - X_n \vert \lt \...

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