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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

A sequence $\{x_n\}$ in an arbitrary metric space, and in particular the space $\Bbb{R}$, is called Cauchy if the terms of the sequence become arbitrarily close together; that is, for every $\epsilon > 0$, there exists an $N$ such that

$$n, m \ge N \implies d(x_n, x_m) < \epsilon$$

where $d$ is the distance function for the metric space. In the particular case of the real numbers, this condition becomes

$$n, m \ge N \implies |x_n - x_m| < \epsilon$$

A complete metric space is a metric space in which every Cauchy sequence is convergent; this gives an alternate definition of convergence of a sequence that does not rely on the limiting value.

Source: the Cauchy sequence article on Wikipedia.

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cauchy sequence integers

let $\{p_n\}$ be a sequence of integers. prove that if $\{p_n\}$ is cauchy, then $\{p_n|n \in N\}$ is finite. my proof: don't know what else do it need or is it right please help... pf: $Given, \{p_n\}$ is sequence of integers which is cauchy. By…
joe
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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 cauchy sequence is finite, won't all the elements…
darkgbm
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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) = \left\{ \begin{array}{ll} 1 & \mbox{if…
Ellya
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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 there general principles I am not aware of that…
NS248
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Having trouble understanding a modular inequality

I am studying this proof for the uncountability of the reals using Cauchy sequences. In a later part of the proof, the author presents a equation that I'm not able to follow. To understand what the terms in the equation mean, I ask that you read the…
Mandar M.
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Does a completion exist in general?

Let $\mathcal H_0$ be a space of functions equipped with an inner product. Is it possible in general to define a “completion” $\mathcal H$ such that $\mathcal H_0 \subset \mathcal H$ and cauchy sequences converge in this space? I thought that it was…
dmh
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Prove that is a cauchy sequence

I have the following problem. Prove by definition that the sequence $r_{n}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...+\frac{(-1)^{n}}{2n+1}$ Is a Cauchy sequence. I think I almost have it, but I have not been able to conclude anything of a…
Haus
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Show that the sequence is Cauchy.

I need to show that $x_n=\frac{1}{n}(1+\frac{1}{4}+\cdots+\frac{1}{3n-2})$ is a Cauchy sequence. For $n \leq m$ , $|x_m-x_n|\leq|\frac{1}{n}(\frac{1}{3n+1}+\frac{1}{3n+4}+\cdots+\frac{1}{3m-2})|$ How should I proceed further?
Nancy
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Prove, that sequence $ b_n = \sin1/2 +\sin2/4 +\dots+ \sin(n)/2^n$ is Cauchy

I do not know, how to start proving that this sequence $$b_n = \sin1/2 + \sin2/4 +\dots+ \sin(n)/2^n$$ is Cauchy. Thanks for all your answers.
Niki Sedlarevič
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If for all $n \in N$, $|a_n| < 2$ and $|a_{n+2} - a_{n+1}| \leq \frac{1}{8}|a_{n+1}^2 - a_{n}^2|$ .

Suppose that ${a_{n}}$ is a sequence such that, for all $n \in N$, $|a_n| < 2$, and $|a_{n+2} - a_{n+1}| \leq \frac{1}{8}|a_{n+1}^2 - a_{n}^2|$ . Prove that ${a_n}$ is a Cauchy sequence. My thought is: First. I know that I will factorize this term…
Intuition
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Is the sequence $a_{n} = 1 + \frac14 + \frac{2^2}{4^2} + \cdots +\frac{n^2}{4^n}$ Cauchy?

I think that it is Cauchy (but I am not sure of this) and this is my proof: $$|a_{m} - a_{n}| = \left|\frac{n+1}{4^{n+1}} + \frac{n+2}{4^{n+2}} + ..... + \frac{m^2}{m}\right| =\sum_{k=n+1}^{m} \frac{k^2}{4^k}$$ And then knowing that $4^n \geq n^2$…
user591668
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Proof of divergence of harmonic series with Cauchy criteria

I have a proof where the divergence harmonic series is shown via the Cauchy criteria. $\epsilon := 1/2$ and $m := 2n$. $$ a_m - a_n = (1 + \frac{1}{2} + \ldots + \frac{1}{m}) - (1+\frac{1}{2} + \ldots + \frac{1}{n}) \\ = (1 + \frac{1}{2} + \ldots +…
Hellstorm
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A property of a Cauchy sequence

Given an injective Cauchy sequence $(x_n)_n$ prove that it has a subsequence $(y_m)_m$ such that $$ d(y_{m+1},y_{n+1}) < d(y_m,y_n)/2 $$ for all $m,n \in \mathbb{N}$. I tried to prove this via induction, namely if we have taken the first n terms we…
Netivolu
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checking a sequence cauchy or not or convergent

Could any one help me to show whether the following are cauchy or convergent? $(1)$ $\{1/n\}$ in $(0,1)$ $(2)$ $\{1/n^p\}$ in $\mathbb{R}, p>0$ $(3)$ $\{n^{\frac{1}{n}}\}$ in $\mathbb{R}$ I loosely understand that $\frac{1}{n}\to 0\notin(0,1)$ and…
Myshkin
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Cauchy sequence problem

Let $(a_n)_{n\in\mathbb N}$ be defined by the recursive formula $a_1 = 1$, $a_{n+1}=\frac{2 + a_n}{1 + a_n}$ for all $n\geqslant 1$. Show that $(a_n)_{n\in\mathbb N}$ is a Cauchy sequence.
user413029
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