Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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55 views

Cauchy sequence such that don't have limes in C[0,1]

Give an example of series $f_n \in C[0,1]$ such that $f_n$ is Cauchy sequence in norm $$\|(a_n)\|_p = \left( \sum_{n=1}^{\infty} |a_n|^p \right)^{1/p}$$ and $$\lim_{n \to \infty} f_n(x)$$ don't ...
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3answers
505 views

Is this space a Hilbert Space?

I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way: $ x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t. $ Is this space a Hilbert ...
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3answers
478 views

Is $x_n=\frac{1}{n}$ Cauchy sequence with the metric $d(x,y)=|\arctan x−\arctan y|$? [closed]

Let $(\Bbb{R},d)$ be a metric space where $d(x,y)=\left\vert\arctan x−\arctan y\right\vert$. Is the sequence $x_n=\frac{1}{n}$ a Cauchy sequence with this metric? The definition of Cauchy sequence ...
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2answers
69 views

Examples of Sequences

Can someone give examples for these types of sequences? A sequence that is monotone but not convergent A sequence that is not bounded but is convergent A sequence that is monotone but not Cauchy A ...
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2answers
91 views

Prove $\sum_{n = 1}^{\infty} 2^{-n} x^n$ does not converge uniformly on $(-2, 2)$

How can one go about proving this? (I understand that the said series does converge uniformly on all $[-a, a]$ where $0 \leq a < 2$.) I am especially interested in knowing if there is a way to ...
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3answers
146 views

Show something is a Cauchy sequence

If I want to show that something is Cauchy, should I show it converges and then show it is Cauchy or should I go at it straight from the definition. I am just trying to figure out generally what to ...
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3answers
104 views

How many real valued Cauchy sequences are there? [closed]

Is the set of all Cauchy sequences of real numbers countable or uncountable? In other words, is $S$ countable or uncountable, where $$S=\big\{\langle x_{n}\vert n\in\mathbb{N}\rangle\in\mathbb{R}^{\...
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4answers
420 views

Prove that $\{n\}$ is a Cauchy sequence that doesn't converge.

Consider the distance function given by $d:\mathbb{R}\times\mathbb{R}\to\mathbb{R},\;d(x,y)=|\int_x^yf(t)dt|$ where $f$ is a continuous and positive function such that $\int_{-\infty}^{+\infty}f$ ...
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1answer
922 views

Prove vectorspace of bounded functions with supremum-norm is complete and no hilbert space

I have the following: Consider the real vectorspace with bounded functions $$V = \{f:[0,1]\rightarrow\mathbb{R} | \exists C > 0 : f([0,1])\subset[-C,C]\}$$ and the supremum-norm $$||f||_\infty =...
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2answers
571 views

What's wrong with the classical Cauchy construction of the reals?

I am reading Bishop's "Constructive Analysis" and he says that defining a real number to just be an equivalence class of Cauchy sequences of rationals would be wrong. Why is that?
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2answers
116 views

What does $\liminf_{n\to \infty} f_n$ and $\limsup_{n\to \infty} f_n$ mean?

I have searched the internet, including this website, and I have found some answers, but none which I understand what actually mean. What is $\liminf_{n\to \infty} f_n$? Is it a single value? Is it a ...
2
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1answer
173 views

On the cauchy sequence $x^n$ in $(C[0,1],\|\cdot\|_\infty)$

I know how to prove that $x^n$ is not cauchy in $(C[0,1],\|\cdot\|_\infty)$, but my question is that since $(C[0,1],\|\cdot\|_\infty)$ is complete, and the pointwise limit on $x^n$ is: $x^n\to f(x) = ...
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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. ...
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1answer
26 views

negative cauchy function

attempt: from defn of uniformally continuous $\forall \epsilon > 0 $ $\exists \delta > 0 $ s.t. $|x-y| < \delta $ with $x,y \in (-\infty,0)$ $\implies |f(x) - f(y)| < \epsilon$. My idea ...
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2answers
518 views

Cauchy sequence and uniform continuity

I read somewhere that because uniform continuous function maps Cauchy sequence to Cauchy sequence and Cauchy sequence is bounded, so the function must be bounded. I am not sure if it is correct. My ...
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2answers
32 views

Construct a sequence with certain property

Construct a sequence $(s_n)$ which satisfies the following property: $\forall x \in \mathbb{R}$ and $\epsilon > 0$, there exists some $N$ such that $|x−s_N| < \epsilon$
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1answer
680 views

Is the set of all integers with metric $d(m,n)=|m-n|$ a complete space?

Consider the set of integers with a metric defined by $d(m,n)=|m-n|$.Is this set complete with respect to this metric? If it is a metric, then I am stuck here. How can a Cauchy sequence have a limit ...
0
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1answer
171 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
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0answers
157 views

Is this sequence a Cauchy sequence?

Consider a continuous mapping $f: X \rightarrow X$, where $X \subset \mathbb{R}^n$ is compact and convex. Consider a sequence $\{x_k \in X\}_{k \geq 0}$ such that for all $k \geq 0$ and $h \geq 1$, $$...
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1answer
4k views

If a sequence converges, then it is Cauchy?

Considering the following proof and its converse: If a sequence converges, then it is Cauchy. That is, if $\lim_{n\to \infty}a_{n} = L$, then given $m>N$, we have that $|a_{m}-a_{n}| < \epsilon$ ...
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2answers
238 views

Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
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2answers
573 views

explore the convergence of series with ln(n)

Help me explore the convergence of red color rounded series. On this photo (the equation below) I used radical indication but it doesn't show me the result. What would be better to use? $$\color{red}{...
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1answer
24 views

Calculating MacLaurin series for $\frac{1}{1-x^2}$

We have the M-series for $\frac{1}{1-x} = \sum_{n=0}^\infty x^n, \frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n,$ and $\frac{1}{1-x^2} = \sum_{n=0}^\infty (x^2)^n$. I need to use the product of the first ...
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2answers
154 views

Showing a sequence is Cauchy

Let $f(x)$ be a function which is differentiable on $\mathbb{R}$ and for which $a = \sup\{|f ′(x)| : x ∈ \mathbb{R}\}$ is less than 1. Let $x_{0}$ be any fixed real number. (a) Show that the sequence ${...
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1answer
64 views

bounded sequence problem

$(a_n)$ is a positive sequence such that: $$\forall \epsilon > 0 ,\quad \exists N\in \mathbb{N}, \quad \forall n,m >N ,\quad |\frac{a_{n}}{a_{m}} - 1|< \epsilon $$ how to prove this ...
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1answer
69 views

Proving Banach's fixed point theorem

The hint tells me how to proceed but I am stuck. I define the sequence ${z_n}$ as is stated in the hint, First off, I want to prove that $|z_{n+k} - z_n| < \epsilon$ I add $z_{n+k-1}$ and ...
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2answers
97 views

Prove ${a_n} = \lim_{n \to \infty } \sum\nolimits_{k \ge 1} {{1 \over {{k^2}}}} $ Converges by using Cauchy's criteria

Prove ${a_n} = \lim\limits_{n \to \infty } \sum\nolimits_{k \ge 1} {{1 \over {{k^2}}}} $ Converges by using Cauchy's criteria. What I did: Let $n, m=n+k \in \mathbb{N}$. $$\left| {{a_m} - {a_n}} \...
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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.
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1answer
30 views

Completeness of special set

Let C denote continuous bounded functions from R -> R which are identically zero outside some closed bounded interval. Is C complete in the sup norm? I think it is not because there will be some ...
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1answer
3k views

Is the space of continuous functions a Cauchy complete?

I am so new to functional analysis so I am looking for an answer of a confusion I am having right now in my mind because I have seen many different answers for the question I am gonna ask below. I ...
3
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1answer
384 views

Cauchy's Theorem and Cauchy's formula

I came across the following problem in our last midterm exam. I am completely stuck as to how to begin the solution: If $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then $|f|$ has no ...
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1answer
2k views

Cauchy sequence of functions and uniform convergence

If $\Omega$ is a bounded domain, and on $C(\bar{\Omega})$ we use the uniform distance $$d(f,g)=\max_{\bar{\Omega}} |f-g|,$$ a Cauchy sequence of functions (w.r.t. the distance $d$) converge and the ...
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1answer
85 views

Example of a sequence with a Cauchy subsequence and terms arbitrarily close

Let $(X,d)$ be a metric space. Suppose there is a sequence $(x_n)$ in $X$ such that $(x_{2n})$ is Cauchy and $d(x_n,x_{n+1})\to 0$. The question is whether $(x_n)$ is Cauchy? It seems intuitively ...
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1answer
78 views

Prove if $\displaystyle \lim_{n \rightarrow \infty} {x_{nk}} = l$ for some $l \in R$, then$\displaystyle \lim_{n \rightarrow \infty} {x_{n}} = l$

Hi guys I know that I should probably be using the Bolzano-Weierstrass theorem and the definition of a Cauchy sequence, but I am not really sure where to get started. Let $(x_n)$ be Cauchy Sequence ...
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4answers
181 views

Completeness proof?

First of all, this is not a question about a specific problem, but more about a general technique. When I face a problem such as "show that a metric space $(M,d)$ is complete", the first thing I do is ...
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1answer
112 views

Is the cube of a Cauchy sequence of real numbers Cauchy?

I am thinking yes, because a Cauchy sequence converges, so we can use the limit law for products twice, declare the cube of the sequence convergent, implying the cube is Cauchy. Is this correct? Are ...
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3answers
1k views

Uniform continuity of a function and cauchy sequences

So I'm pretty sure this is almost immediate from the definitions, please tell me if I am incorrect.. Consider two cauchy sequences in D, $\{x_n\}$ and $\{y_m\}$. Since $f$ is uniform continuous we ...
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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([...
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1answer
967 views

Question concerning the mean-value property

If $U$ is an open subset of $\mathbb C$, the mean square norm is defined as: $$||f||_{L^2 (U)} = {\left(\int_U |f(z)|^2 dxdy\right)}^{1/2}$$ And the norm supremum is defined as: $$||f||_{L^{\infty} ...
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1answer
76 views

Proof that this sequence is Cauchy

How to prove that if ${a_n}$ is a Cauchy sequence, if $x_n$ is a sequence and there is a constant $C$ such that, $$|x_n-x_m| \le C|a_n-a_m|$$ Then $x_n$ is also a Cauchy sequence. Any hints or ...
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3answers
613 views

Proving that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.

Let $(x_n)_{n=1}^{\infty}$ be a sequence such that $|x_{n+1} - x_n| < r^n$ for all $n \geq 1$, for some $0 < r < 1$. Prove that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence. I understand that ...
4
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3answers
1k views

Find a limit of the recursive sequence

The task is to prove sequence convergence and find a limit. $x_0=0$ $x_1=1$ $x_{n+1}=\frac {x_n + n \cdot x_{n-1}} {n+1}$ I have computed some values of a sequence to build up some idea of the data: ...
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1answer
277 views

Proof That a Sequence is Cauchy [duplicate]

$\{a_n\}$ is a sequence such that $|a_{n+1} - a_n| < 5^{-n}$ for all $n>0$. How to prove that this sequence is a Cauchy sequence?
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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 $\rho$ ...
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1answer
131 views

Proof check for completeness

I'd like to see if the proof I have is adequate. Statement. Let $X$ and $Y$ be Banach space, the product $X\times Y$ is a vector space under coordinate operations with norm $$ \|(x,y)\| = \|x\|_X +\|...
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1answer
73 views

Fixed point question with convergence

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ is $C^1$ and $1$ to $1$ and there exists a strict increasing sequence $t_{n} \in \mathbb{N}$ s.t $f^{t_{n}}(x) \to p$ for all $x$ as $n\to \infty$ (composition $...
2
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1answer
41 views

$f$ extends continuously to the completion $(\bar{X},\bar{d})$

I have to prove of disprove the following fact: Let $(X,d)$ be a metric space and $f: X \rightarrow \mathbb{R}$ be a continuous function. Then $f$ extends continuously to the completion $(\bar{X},\...
2
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0answers
55 views

Show, using only the definition, that if $\{X_n\}$ is Cauchy and $C\in\mathbb{R}$ then $\{CX_n\}$ is Cauchy.

Let me know if what I did this correct please. Let $\epsilon>0$ be given, we want to find $N\in\mathbb{N}$ such that $|CX_n-CX_m |<\epsilon$ $\forall n,m\geq N$. $$|CX_n-CX_m |=|C||X_n-X_m |$$ ...
2
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1answer
372 views

Cauchy sequence $a_n = 1 + \frac12 + \frac14 + … + \frac{1}{2^n}$

For the sequence $a_n = 1 + \frac12 + \frac14 + ... + \frac{1}{2^n}$, $n \ge 1$, find a formula $N = N(\epsilon)$ such that for all $\epsilon > 0$ and for all $m,n \ge N(\epsilon)$, $|a_m - a_n| &...
1
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2answers
179 views

Laurent Series, Cauchy, Pole Order

http://gyazo.com/8ef04b854bc3bbfb6b55a9af45e51fdc.png Since $f(z)$ not differentiable at $0$, isolated singularity at $z = 0$. By expanding the Laurent series and looking at the first term, I got a ...