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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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1answer
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Given a sequence in $\mathbb{R}$ is it sufficient to prove $\lim d(x_n,x_{n+p}) =0$ to show that it is cauchy? [on hold]

I mean if i first take limit $p$ tending to infinity and then take limit $n$ tending to infinity.If the limit turns out to be $0$ can i conclude that sequence is cauchy?
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1answer
20 views

Proof completenes of $ \{x \in \mathbb{C}^\mathbb{N}\ |\ \sum_{n=1}^\infty s_n |x_n|^p < \infty \}$

Let $(s_n)_{n\in\mathbb{N}} \subseteq \mathbb{R}$ such that for all $n$: $0 < s_n \leq \frac{1}{n} $. Let $p>1$. How to show that the space of sequences $ l^p_s := \{x \in \mathbb{C}^\mathbb{...
1
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1answer
65 views

Prove convergence and find limit of recursive sequence [duplicate]

Letting $a_{1}=2$ I have a recursive sequence defined as follows: $$ a_{n+1} = \frac{a_{n}}{2} + \frac{5}{a_{n}} \ \ \forall n \geq1$$ How can I prove that the sequence {$a_{n}$} converges and also ...
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2answers
40 views

Limit of $a_n=(1-\frac13)^2\cdot(1-\frac16)^2\ldots(1-\frac{1}{\frac{(n)(n+1)}{2}})^2 \; \;\forall n \geq 2$

$$a_n = \left(1-\frac13\right)^2\cdot\left(1-\frac16\right)^2\ldots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2 \; \;\forall n \geq 2$$ I have no idea how to solve this, however I will give it a try ...
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2answers
39 views

Find a Cauchy sequence that doesn't $p$-converge to any rational number.

Let $p$ be a prime number. For any ratinoal number $x$, define $$|x|_p = \begin{cases} 0 \,, & \mbox{if } \,x=0 \\ p^{-\alpha}\,, & \mbox{if }\,x=p^\alpha\frac{n}{m} \,\,,\mbox{in which }m,n\...
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1answer
41 views

Prove that if $\sum_{n = 1}^{\infty} a_{n}$ converges, then $\sum_{k = 1}^{\infty} A_{k}$ converges(2) [duplicate]

Suppose that $\{a_n\}$ be a sequence of real numbers.let $\{n_{k}\}$ be an increasing sequence of positive integers, and let $A_{k} = a_{n_{k}} + a_{n_{k}+1}+ ... +a_{n_{k +1}-1}$, for each $k \in \...
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1answer
37 views

Showing $\{f(\frac{1}{n+1})\}$ converges in $\mathbb{R}$

Question: Let $f:(0,1) \to \mathbb{R}$ be a differentiable function such that $|f'(x)| \leq 5$, for all $x \in (0,1)$. Show that the sequence $\{f(\frac{1}{n+1})\}$ converges in $\mathbb{R}$. ...
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2answers
37 views

Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous [duplicate]

Does there exists a function which maps Cauchy sequence to Cauchy sequence but it isn't uniformly continuous? we know uniformorly continuous function maps cauchy sequence to cauchy sequence.But my ...
2
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1answer
48 views

If $\{a_n\}$ is bounded and non-decreasing, prove that $\liminf b_n = 0$, $b_n = n(a_{n+1} - a_{n})$

Let $\{a_n\}$ be a bounded and non-decreasing sequence of reals, and $b_n = n(a_{n+1} - a_{n})$. (a) Show that $\liminf b_n = 0$ (b) Give an example of a sequence $\{a_n\}$ such that $\{b_n\}$ ...
3
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2answers
45 views

Existence of $a_k$ such that $\sum_k a_kb_k<\infty$ and $\sum_k a_k=\infty$ given $b_k\to 0$

I was working with a problem from functional analysis. I reduced the problem to the following problem: Let $b_k>0$ be a decreasing sequence converging to $0$. Does there exist a non-negative ...
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2answers
76 views

find the point of convergence of sequence {$a_n$} [duplicate]

Let $\displaystyle a_n= \sum_{k=1}^{n} \frac{n}{n^2+k}$, for $n\in \mathbb{N}$. Then what is the nature of sequence $\{a_n\}_{n\in\mathbb{N}}$. I tried using the Cauchy's general principle of ...
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1answer
25 views

Limit of sequence of sequences

I got to thinking about sequences of Cauchy sequences. Here is a simple example. Let us define $b_n = (n,1,1,1,...)$ for $n\in\mathbb N$. So we have \begin{align} b_1&=(1,1,1,1,...)\\ b_2&=(2,...
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2answers
50 views

For a sequence $(a_n)$ of real numbers, $\sum_{n=1}^\infty |a_{n+1}-a_n|$ converges implies $(a_n)_{n\in\mathbb{N}}$ converges.

I know that when the series $\sum_{n=1}^\infty |a_{n+1}-a_n|$ converges, then we have $|a_{n+1}-a_n|\rightarrow 0$ So by using this I was going to prove that the sequence $a_n$ is Cauchy. But couldn't ...
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1answer
29 views

Sequences of sequences: question about Cauchy's construction of the real numbers

As is well known, one way of constructing the real numbers is to consider Cauchy sequences and call two of them equivalent if they have the same limit. I got to thinking about the Cauchy sequences ...
0
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1answer
21 views

Confusion regarding Cauchy's General Principle and Uniform Convergence

The definitions of the two are so alike, that it confuses me. Cauchy's General Principle: The necessary and sufficient condition that a function $f(x)$ may tend to a definite limit, say $l$, as $x ...
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2answers
56 views

Prove this sequence converges or diverges to $-\infty$

Let $a_n$ be a sequence such that for every $n$: $a_n\le\frac{1}{2}(a_{n-1}+a_{n-2})$. Prove that $a_n$ either converges to a real number $L$ or diverges to $-\infty$ $(L\in[-\infty,\infty))$. ...
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1answer
22 views

Comparison Test for complex series: logical argument?

I am trying to show that the comparison test holds for complex series, meaning: if $\sum_{n=0}^{\infty} z_n $ is a complex series and $\sum_{n=0}^{\infty} a_n $ is a convergent non-negative real ...
0
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1answer
44 views

Convergence of a sequence defined by a recurrence inequality [duplicate]

Let $(u_n)$ be a sequence of real numbers with the following property \begin{eqnarray*} \forall\,m,n\ge1,\quad u_{m+n}\le \frac{m}{m+n}u_m+\frac{n}{m+n}u_n \end{eqnarray*} Is it true that $(u_n)$ ...
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2answers
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Meaning of being a Cauchy sequence

A Cauchy sequence in a metric space $(X,d)$ is a sequence for which the distance between two terms can be made as small as we want, provided we look far enough in the sequence. Let $X \subseteq Y$, ...
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2answers
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Please help me to understand a proof that $a_n = \frac{\tan 1}{2} + \frac{\tan2 }{2^2} + \dots + \frac{\tan n}{2^n}$ is a Cauchy sequence

$\textbf{Problem:}$ Show that $$a_n = \frac{\tan 1}{2} + \frac{\tan2 }{2^2} + \dots + \frac{\tan n}{2^n}$$ is a Cauchy sequence. This is the solution. Here's my question: In the first line, we have $...
6
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1answer
126 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 ...
3
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3answers
70 views

Proof verification. $\{x_n\}$ is a sequence such that $|x_{n+1} - x_n| \le C\alpha^n$ for $\alpha\in (0, 1), n\in\Bbb N$. Prove $x_n$ converges.

Let $\{x_n\}, n\in \Bbb N$ denote a sequence such that: $$ \begin{cases} |x_{n+1} - x_n| \le C\alpha^n \\ 0 < \alpha < 1 \end{cases} $$ Prove $\{x_n\}$ converges. Given the fact $|x_{n+1} - ...
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1answer
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Sequence of function on $\mathbb{R}$ Cauchy iff convergent

Theorem: Let $(f_n)$ be a sequence of functions on $I \subseteq \mathbb{R}$. Then $(f_n)$ pointwise convergent iff pointwise cauchy. Here, I only prove "$\Longleftarrow$" since the converse is very ...
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0answers
59 views

Proof verification of $x_n = {1\over2^2}+{2\over3^2}+\cdots+{n\over(n+1)^2}$ diverges by the negation of Cauchy

Let $\{x_n\}$ denote a sequence: $$ x_n = {1\over2^2}+{2\over3^2}+\cdots+{n\over(n+1)^2} $$ Show that $\{x_n\}$ diverges using the negation of Cauchy criterion. I would like to kindly request a ...
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1answer
41 views

Prove that $\{x_n\}$ is a Cauchy sequence iff $\forall \epsilon > 0\ \exists N: \forall n > N \implies |x_n - x_N| < \epsilon$

Let $\{x_n\}$ denote a sequence. Prove that: $$ \{x_n\}\ \text{is fundamental} \iff \forall \epsilon > 0\ \exists N: \forall n > N \implies |x_n - x_N| < \epsilon $$ Let $P$ be a ...
5
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1answer
76 views

Is the notion of Cauchy sequences definable in a bornological topological space?

Being a Cauchy sequence is not a topological property, i.e. two metrics can induce the same topology and yet a sequence which is Cauchy in one may not be Cauchy in the other. It is a uniform property ...
2
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1answer
28 views

Proof verification of $x_n = \sum_{k=1}^n a_kq^k$ is Cauchy given $|a_k| \le C, |q| < 1, k\in\Bbb N$

Given a sequence $\{x_n\}$: $$ x_n = \sum_{k=1}^n a_kq^k $$ and: $$ \begin{cases} |a_k| \le C\\ |q| < 1\\ k\in\Bbb N \end{cases} $$ Prove $\{x_n\}$ is a fundamental sequence. By ...
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2answers
57 views

Proof verification that $\{x_n\} = 0,\underbrace{77\dots 7}_{\text{n times}}$ is a Cauchy sequence.

Given a sequence $\{x_n\}$: $$ x_n = 0,\underbrace{77\dots 7}_{\text n\ times} $$ Prove that $\{x_n\}$ is a Cauchy sequence. Recall the definition of a fundamental sequence: $$ x_n\ \text{is ...
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2answers
26 views

Identical Cauchy sequences and continuity.

Find a set $X$ and two metrics $d$ and $m$ on $X$ such that the Cauchy sequences of $(X,d)$ and $(X,m)$ are identical and the identity map from $(X,d)$ to $(X,m)$ is continuous but not uniformly ...
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0answers
50 views

Proving that every Cauchy sequence in measure converges in measure

Let $(X,\mathcal{A},\mu)$ be a measure space and $(f_n)$ a sequence of real-valued functions on $X$ which is Cauchy in measure; that is, for any $\epsilon>0$ there exists $N\in\mathbb{N}$ such that ...
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3answers
26 views

Show that if two metrics induce the same topology, one metric space is compact iff the other one is.

Question: Given two metric spaces $(X, d)$ and $(X,e)$, where $\tau(X,d)$ = $\tau(X,e)$, show that $(X,e)$ is compact if and only if $(X,d)$ is compact. I'm aiming to prove this using the definition ...
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2answers
63 views

How to show that $\sin(n)$ does not converge ONLY by using Cauchy's criterion?

I know this question has been asked before... I went through all of the questions of this sort and none of them had an answer using Cauchy's criterion. I know that $\sin(n)$ does not converge and I ...
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2answers
51 views

Characterization of the sequence $x_1=\cos(x), x_{n+1}=\cos(x_n)$ where $x>0.$

Let $x>0$, $x_1=\cos(x),$ and $x_{n+1}=\cos(x_n), \forall n\geq1.$ Then the sequence $(x_n)_{n\geq 1}$ is bounded but not monotone, bounded but not Cauchy, Cauchy, convergent. It ...
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1answer
29 views

Prove that $\{x^n\}$ is Cauchy in $S\subseteq \ell_\infty$

I'm kind of new into Functional analysis. So, I have the following question bothering me. Let $S$ be the set of sequences having only a finite number of non-zero terms. Clearly, $S\subseteq \ell_\...
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1answer
15 views

Show that there exists a measurable set $A\in\mathcal{A}$ such that $\chi_{A_n}\to\chi_{A}$ $\mu$-a.e.

I'm studying for measure theory and I am wondering how to understand a line in the proof, as well as how the problem could be solved without referencing the fact that $L_p$ spaces are Banach: ...
0
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1answer
21 views

Prove this is a cauchy sequence [duplicate]

Let ${a_n}$ be a sequence such that there exists an M > 0 such that for all n ∈ N one has $|a_{n+1} − a_n|$ ≤ M/$2^n$ Prove that ${a_n}$ is a Cauchy sequence. My attempt: I tried to use the triangle ...
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2answers
53 views

Cauchy´s convergence test for Series

Show whith the cauchy's convergence test for series, that the sequence : $$ b_n = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} $$ converges. I think it would be not so difficult to ...
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2answers
197 views

Determine if this specific sequence is a Cauchy sequence

I have the following sequence: $$a_n =\sum_{k = 1}^n (-1)^{b_k} {1\over k^2} $$ And the hint is that I have to prove that: $$ {1\over k^2} < {1\over k-1} - {1\over k} $$ So assuming $m>n$, I ...
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1answer
42 views

Show that $ \lim_{n\to\infty} \int_{0}^{1} f_n $ exists

Let $(X,d)= (C[0,1],d)$ where $C[0,1]$ is the set of real-valued continuous functions on $[0,1]$ and $d= \int_{0}^{1} |f-g|$ is the Riemann Integral. Suppose $(f_n)$ is a Cauchy sequence in $(X,d) $ ...
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0answers
14 views

Is there a space where no convergent Cauchy sequences exist?

I understand every non-complete space can be made complete by including the limit points of convergence of Cauchy sequences. I assume that the opposite process is possible as well. But, do spaces ...
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2answers
38 views

Prove that $(X,\Vert {\cdot}\Vert_2)$ is not complete where $\Vert f\Vert_2=\left(\int_{-2}^{2}|f(t)|^2 dt\right)^{1/2}$ [duplicate]

Prove that $(X,\Vert {\cdot}\Vert_2)$ is not complete where $X=C[-2,2]$ and \begin{align}\Vert f\Vert_2=\left(\int_{-2}^{2}|f(t)|^2 dt\right)^{1/2}.\end{align} MY TRIAL It suffices to produce a ...
0
votes
3answers
67 views

Cauchy sequence and subsequence [closed]

"Let $\{x_n\}\subset U$ be a Cauchy sequence. Give a direct proof that if a subsequence $\{x_{nk}\}\subset S$ has a limit $L$, then the Cauchy sequence $\{x_n\}\subset U$ has $L$ as a limit. Do not ...
4
votes
3answers
58 views

Showing sequence is Cauchy by Definition

Question: I have to show that sequence $(x_n)$ defined by $x_n=\frac{n+(-1)^n}{2n-1}$ , $n=1,2,3,...$. Is Cauchy sequence using definition only. My attempt: (I can see given sequence is Cauchy ...
1
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2answers
45 views

Show that $|\frac{1}{2n}-\frac{1}{2m}| < \epsilon$ holds for all $m, n > \frac{1}{\epsilon}.$

In Example 1.5-9 of the book Functional Analysis by Kreyszig it claims that $|\frac{1}{2n}-\frac{1}{2m}| < \epsilon$ holds for all $m, n > \frac{1}{\epsilon}.$ My calculations don't lead to ...
2
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2answers
34 views

Prove a closed subset of a complete metric space is complete via contradiction

I've seen some proofs by definition and just want to ask for proof verification on whether this is okay as well: Given complete metric space $(X,d)$ and $A \subset X$, $A$ closed. Prove $A$ is ...
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1answer
50 views

Continuous linear operator preserving Cauchy sequence in metric vector spaces

Let $T: X \rightarrow Y$ be a continuous linear operator. $(X,\rho), (Y,\xi)$ are linear metric spaces and $\{x_n\} \subset X$ is a Cauchy sequence. I need to show that $\{Tx_n\}$ is a Cauchy ...
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 ...
0
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1answer
32 views

Is $T$ bounded if $(X,d)$ is not complete?

I) Let $b_1,b_2,...,b_n....$ be a Cauchy sequence in a complete metric space $(X,d)$. If $T$ is the set of points in the sequence show that $T$ is a bounded set. As the space is complete, every ...
1
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1answer
35 views

Banach space with subset whose elements are at least $d\gt 0$ far from each other is not separable

Let $X$ be a Banach space, and $A\subseteq X$ subgroup, where $A$ is not countable, and there is some $d \gt 0$ such that for all $x,y \in A$: $||x-y||>d$. Prove that $X$ is not separable. My ...
0
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6answers
1k views

A sequence in $\mathbb{R}$ that has no Cauchy subsequence

Give an example of a sequence in $\mathbb{R}$ which has no subsequence which is a Cauchy sequence. I can find out a sequence that is not a Cauchy sequence such as $\{\ln(n)\}$ once $|\ln(n)-\ln(n+1)|=...