Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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71 views

set of Cauchy sequences complete matrix

Given a set S, let S* be the set of all Cauchy sequences. Is it true that S* is a complete metric space? Suppose that $A, B \in S^*$. Then the metric is $\rho(A,B) = \lim_{v \to \infty} \rho(a_{v},b_{...
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725 views

Prove sequence is Cauchy.

Prove the sequence $\{a_i\}$ defined by $a_1=1 \text{ and } a_{i+1} = 1 + \frac{1}{a_i}$ is Cauchy. And prove it converges to $\sqrt{2}$. I want to show $\lim\limits_{n\to\infty}(a_{i+1}-a_i)=0$ for ...
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1answer
211 views

Why isn't $\,\mathcal C[0,1]$ a Banach space in this unusual norm?

I wish to ask the following question: Let $\mathcal X$ be the normed space $\,\mathcal X=\mathcal C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't $\...
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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 ...
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485 views

Finite Cauchy Sequence

How do you prove that given a sequence that is finite, the sequence always converges? Or, must the sequence be cauchy for this finite sequence to converge? How can a cauchy sequence be finite? If a ...
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691 views

Sum of the distance between consecutive terms implies the sequence is Cauchy.

If a sequence $(x_n)_{n=1}^{\infty}$ in $\mathbb{R}^n$ satisfies $\sum_{n\geq 1} ||x_n-x_{n+1}||<\infty$, show that it is Cauchy. This isn't a complete answer, but here's my train of thought. ...
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1answer
56 views

Cauchy Sequences Lemma in Vector Space E

I ran into a Lemma. Suppose $||.||_1$ and $||.||_2$ are two norms in vector spapce E, such that $||.||_1$ and $||.||_2$ are equivalent norms and {$x_n$} is an equivalent in E, then {$x_n$} is ...
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3answers
71 views

Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
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If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is continuous , is $f(x_n)$ a cauchy sequence?

If $X = \{x_n:n \in \mathbb N\}$ is a cauchy sequence in a metric space $S$ and $f : S \rightarrow T$ is a continuous function where $T$ is an another metric space , is $f(x_n)$ a cauchy sequence? ...
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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 ...
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930 views

If $a_{n+1}=a_n+\frac1{a_n}$, then $a_n/n$ converges to $0$

Let $a_{n+1}=a_n+\dfrac1{a_n}$, with $a_n=1$. Prove $\lim \limits_{n\to \infty }\left(\dfrac{a_n}{n}\right)=0$. Now I already know that it is monotonically increasing and that $a_n\to \infty$ as $n\...
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1answer
928 views

Cauchy sequences. Show that $(x_n)$ is Cauchy.

Let $(x_n)$ and $(y_n)$ be sequences such that $\lim y_n = 0$. Suppose that for all $k \in \Bbb N$ and all $m ≥ k$ we have $|x_m − x_k| ≤ y_k$. Show that $(x_n)$ is Cauchy. I need a little guidance ...
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3answers
311 views

Is the sequence $x_n = \frac{\sin 1}{2} + \frac{\sin 2}{2^2} + \frac{\sin 3}{2^3} +\dots + \frac{\sin n}{2^n}$ Cauchy?

$$x_{n} = \frac{\sin 1}{2} + \frac{\sin 2}{2^2} + \frac{\sin 3}{2^3} + ... + \frac{\sin n}{2^n}$$ I came across this sequence while studying, and while it is convergent, I'm curious as to whether or ...
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714 views

Constructive proof for existence of integer part of real number

I try to prove de following exercise of my analysis textbook. Show that for every real number $x$ there is exactly one integer $N$ such that $N \le x < N + 1$. I have been finding a ...
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4answers
195 views

Show a sequence such that $\lim_{\ N \to \infty} \sum_{n=1}^{N} \lvert a_n-a_{n+1}\rvert< \infty$, is Cauchy

Attempt. Rewriting this we have, $$\sum_{n=1}^{\infty} \lvert a_n-a_{n+1}\rvert< \infty \,\,\,\Longrightarrow\,\,\, \exists N \in \mathbb{N}\ \ s.t,\ \ \sum_{n \geq N}^{\infty} \lvert a_n-a_{n+1}\...
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1answer
77 views

Cauchy sequence and metrics

I'm having trouble with another analysis homework problem: Let $x_n$ be a sequence in $\mathbb{R}$ such that $d(x_n, x_{n+1}) \le \frac{d(x_{n-1},x_n)}{2}$. Show that $x_n$ is a Cauchy sequence. I ...
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How to know if a space has a convergent subsequence?

So this is something I have been struggling with lately... how do we generally know that a space/set has a subsequence that converges? The current one I am struggling with is the space of sequences ...
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92 views

Prove that $d_n$ is a Cauchy sequence in $\mathbb{R}$

Let $(x_n$) and $(y_n)$ be Cauchy sequences in $\mathbb{R}^n$ , i.e. lim$_{n,m}$ |$x_n$ − $x_m$| = $0$ and lim$_{n,m}$ |$y_n$ − $y_m$| = $0$. For each n, let $d_n = |x_n − y_n|$. Prove that $d_n$ is a ...
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3answers
600 views

How to prove the sequence $x_{n+1} = \frac{x_n}2 + \frac 1{x_n}$ is a Cauchy sequence

Here is a question I do not know how to prove. Thanks for your helping! Prove that the sequence $$x_{n+1} = \dfrac{x_n}2 + \dfrac 1{x_n}, x_0 = 1$$is a Cauchy sequence.
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183 views

Convergence of a Cauchy sequence

I want to prove that $x_n \to 0$ given the following: $x_n$ is a Cauchy sequence and for all $\epsilon > 0 $ there exists a $n > \frac{1}{\epsilon}$ (let's call this number $n_1$) such that $|...
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1answer
316 views

Find a Cauchy sequence that does not converge

I am supposed to look at $l_0$, the set of all sequences with finitely many non-real elements in $(l_0,d_{\infty})$. It is just that I don't quite understand how the $d_\infty$-metric is defined on ...
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2answers
425 views

Prove that the space of sequences under this metric is complete and compact.

I'm currently studying for the prelim exams, and I would love a hint on how to complete this problem. If $X$ is the space of sequences of $0$'s and $1$'s (i.e., $x \in X$ if $x = (x_{1}, x_{2}, x_{...
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3answers
4k views

Prove a Sequence is Cauchy

Prove that sequence {(2n+1)/n} is Cauchy. I understand the definition of a Cauchy sequence; however, I'm not sure how to find the necessary value of N to satisfy the prove. I know that you can ...
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168 views

Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
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99 views

convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
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4answers
953 views

Why are all convergent sequences necessarily Cauchy?

I can understand the proof, which I could do myself: $|s_n - s_m| = |s_n - s + s - s_m|$ $\Rightarrow |s_n - s_m| \leq |s_n - s| + |s_m - s| $ For some $\epsilon > 0, \exists\ \ N(\epsilon) \...
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1answer
249 views

Cauchy sequences are bounded in every metric space

A few days laid out an example, and asked for help, and @ shadow10 replied, his answer the question of can I find the https://math.stackexchange.com/questions/879662/every-Cauchy-sequence-is-...
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1answer
44 views

Prove that for any $\{x_{m_k} \}\in I_n$, where $I_n$ are dyadic intervals, $\lim_{n \to \infty} x_{m_k} =c$

Proving for any $\{x_{m_k} \}\in I_n$ that:$$\lim_{n \to \infty}\{x_{m_k} \}=c$$ I have been trying to solve this problem, but i dont know how to write it properly, so i need your help whit writing ...
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1answer
477 views

Sequence in $\mathbb R^2$ converges if and only if it is Cauchy

How to prove that a sequence $(x_n)$ in $\mathbb R^2$ converges if and only if it is Cauchy? I've proven that the triangle inequality holds for the euclidean norm of vectors.
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1answer
133 views

How to prove the 'uniform summability' of a Cauchy sequence?

I have an exercise given by the teacher and I'm pretty sure that this proof is not hard, but I don't have idea how to approach it. I have to prove the 'uniform summability' (this name was used by ...
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1answer
661 views

Convergence of a Cauchy sequence of matrices

I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta \...
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1answer
601 views

Cauchy sequence of random variable

I know (from calculus) the Cauchy convergence theorem for sequence of real numbers...how to show it with random variables? I have $S_n=\sum_{i=1}^n X_i;$ $X_i$ being iid centered r.v., $S_n$ is a ...
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4answers
169 views

Show that $\langle f_n \rangle$ is a Cauchy sequence, where $f_n=1-\frac12+\frac13-\frac14+\dots+\frac{(-1)^{n-1}}{n}$

Show that $\langle f_n \rangle$, where $$f_n=1-1/2+1/3-1/4+\dots+\frac{(-1)^{n-1}}{n}$$ is a Cauchy sequence. My attempt: Consider $$|f_{2m}-f_m| = \left| \frac{(-1)^m}{m+1}+\frac{(-1)^{m+1}}{m+2}+\...
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2answers
304 views

Does a bounded continuous function map Cauchy sequences to Cauchy sequences?

I only ever see the example of $f:(0,1]\rightarrow \mathbb{R}$ where $f(x)=\frac{1}{x}$as that of a continuous function that does not map Cauchy sequences to Cauchy sequences. Are there examples of ...
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4answers
527 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 $f_n$ ...
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a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [closed]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
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418 views

Negation - Cauchy sequence

Suppose that $(x_i)_{i \geq 1}$ is not a Cauchy sequence of real numbers. How to prove that there exist $\varepsilon >0 $ and an increasing sequence $(i_n)$ of indices such that $$ |x_{i_{n+1}}-x_{...
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1answer
64 views

Proof sequence is cauchy

Prove that a sequence satisfying: $|x_k-x_{k+1}|< \frac{1}{a^k}$ for some $a>1$, for all $k$ is cauchy. It follows from the criterion that: $|a_n - a_m|<\sum_{i=n}^{m}{\frac{1}{a^i}}, m&...
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3answers
411 views

$\{a_n\}$ is a Cauchy sequence, if $a_{n+2} = \frac{a_n + a_{n+1}}{2}$

Suppose that the sequence $\{a_n\}$ satisies the relation $$ a_{n+2} = \frac{a_n + a_{n+1}}{2}, $$ for all $n \in \mathbb{N}_{+}$ Prove that $\{a_n\}$ is a Cauchy sequence ...
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1answer
149 views

Prove $a_{n+1} = \frac{4+3 a_n}{3+2 a_n}$ is a Cauchy sequence [closed]

How to prove that a sequence $a_n$ as defined $a_{n+1} = \frac{4+3 a_n}{3+2 a_n}$ is a Cauchy sequence?
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2answers
432 views

Sequence of bounded linear operators implicating Cauchy sequence in $\mathbb K$

Let H be a Hilbert space and $(T_n)_{n \in \mathbb N}$ be a sequence in ${\rm BL}(H)$ (bounded linear operators) such that $(\langle y,T_nx \rangle)_{n \in \mathbb N}$ is a Cauchy sequence in $\mathbb ...
3
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1answer
609 views

Limit of $x_n = 0.5(x_{n-1} + x_{n-2})$ - help finishing proof…

EDIT: Fixed geometric proof off-by-one error. I am looking for the limit of $x_n = 0.5(x_{n-1} + x_{n-2})$ with $x_2 > x_1$ arbitrary. I can show $\forall n: |x_{n}-x_{n+1}| = \frac{c}{2^{n-1}}$ (...
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1answer
3k views

Completeness of the set of convergent sequences

It's a problem from the book "Topology of Metric Spaces", written by Kumaresan: "Show that the set $\textbf{c}$ of convergent sequences in the Normed Linear Space of all bounded real sequences ...
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44 views

If $\{X_n\}$ is a Cauchy seq of r.vs a.s., then why does it converge to some r.v?

I was suddenly suspicious of what title says. My logics are follows. $\{X_n\}$ is a Cauchy seq of r.vs $P$-a.s. $\Leftrightarrow$ $P(\{X_n\}$is a Cauchy $)=1$ $\Leftrightarrow$ $\exists ...
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1answer
153 views

Determine whether $f_n(x) = \frac {nx}{1+(nx)^2}$ is cauchy in $[ C^0([−1, 1], \mathbb {R} ), d_\infty]$

I have a homework question: Is the sequence $$ f_n(x) = \frac {nx}{1+(nx)^2} $$ Cauchy in the space $ C^0([−1, 1], \mathbb {R} ) $ with the metric induced from the sup norm? Could you please write ...
3
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1answer
112 views

Can I take Inverse Limits as Cauchy sequences literally?

I have been told to think of inverse limits as "Cauchy Completions" under some metric, for instance through the construction of the p-adic numbers. This got me thinking, though, and I wonder if the ...
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3answers
123 views

Let $a_n=\frac{a_{n-1}+a_{n-2}}{2}$ for each positive integer $n\geq 2$. Show that $\{a_n\}_{n=1}^{\infty}$ is Cauchy

Let $a_0$ and $a_1$ be distinct real numbers. Define $a_n=\frac{a_{n-1}+a_{n-2}}{2}$ for each positive integer $n\geq 2$. Show that $\{a_n\}_{n=1}^{\infty}$ is a Cauchy sequence. Hint: You may want to ...
1
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1answer
139 views

Completeness: Nets vs. Sequences [duplicate]

Prove that a metric space is complete w.r.t. sequences iff it is complete w.r.t. nets! (The converse is trivial of course!)
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2answers
53 views

Convergence of series $\sum a_rb_r$

Assuming $\sum a^2_r$ and $\sum b^2_r$ converge, can we deduce that $\sum a_rb_r$ converges? It feels like we can, but how? Using Cauchy Criterion for convergence maybe? Can you hint me? Thanks a ...
2
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0answers
331 views

Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the function $f_n$ and show that V is not a Banach Space.

The Assignment: Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the continuous function $f$ for all $n \in \mathbb{N}$: $$f_n: [0,1] \rightarrow \mathbb{R}, \ x\...