Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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1answer
59 views

Do Cauchy always converge (for some superset)?

I know Cauchy sequences are only guaranteed converge in complete metric spaces. However, I have been struggling with one issue. It seems that every Cauchy sequence converges, at least in a larger ...
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1answer
40 views

On proving that a certain sequence is Cauchy in a Hilbert space.

Suppose we have an orthogonal family $\,\phi _{n}~~, n=1,2,...,$ in a Hilbert space, and $a_{n}$ is any sequence of real scalars such that $\,\sum_{i=1}^{\infty}\left| a_{i}\right|^{2}\langle\phi _{i},...
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1answer
109 views

When does $d(x_n,x_{n+1})\rightarrow 0$ imply $(x_n)$ is a Cauchy sequence?

http://math.stackexchange.com/q/107336/21436 answers the question as to wether or not this "weaker" Cauchy criteria is equivalent to the actual cauchy criteria. I've seen the examples of why that's ...
4
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3answers
139 views

To show that a recursively defined sequence by $x_1=\frac12$ and $x_{n+1}=\frac{x_n^3 + 2}{7}$ is Cauchy - How?

Show that the sequence defined by $x_1$ = $\frac{1}{2}$ and $x_{n+1} = \frac{x_n^3 + 2}{7}$ for $n \in N$ satisfies the Cauchy criterion. I don't understand how to go about this problem. Is it ...
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2answers
36 views

Problem with showing that every Contractive Sequence is Cauchy

Here's the proof which shows that every contractive sequence satisfies the Cauchy criterion, and I don't seem to be convinced with it. My doubts: Why is the fact that the RHS tends to zero on ...
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0answers
43 views

Prove following sequence is a Cauchy sequence. [duplicate]

Let $ (a_{n})_{n \in \mathbb{R}} $ be a recursive sequence and $a_{n+1} - a_n = \frac{1}{2}(a_{n-1}-a_n)$. I have to proof that this a Cauchy sequence and I have to calculate te limit given that $a_0 ...
5
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2answers
119 views

Intuition for non-convergence of Cauchy sequence in $\mathbb{Q}$

Suppose we were standing on the rational line at the point 3. Then we took a step to the point 3.1, then to 3.14, etc. (Cauchy sequence of decimal approximations of $\pi$). Suppose, also, that it ...
2
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0answers
40 views

Convergence of complex sequence with gradient descent

I am looking to solve the following otimization: $\underset{{{{\bf x} \in \mathcal{X}}}}{\text{min}} \; f({\bf x})$ where ${\bf x} \in \mathbb{C}^N$ and $f({\bf x}) \in \mathbb{R}$, $f(x)$ is a ...
0
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2answers
63 views

Help needed: Limit of the sequence $s_n = \frac{1}{n}\sum a_i$ [duplicate]

We define $ {a_{n} } $ as a convergent sequence converging to a real number , say $ a $ then we set to define another sequence $ {s_{n}} $ such that it's terms are defined as $ s_{n} = \frac {1}{n} \...
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3answers
53 views

Hard time understanding Cauchy criterion and the convergence criterion using a Cauchy series as an example

$$b_n=\frac{x}{1}+\frac{x+1}{3}+\frac{x+2}{5}+\frac{x+3}{7}...+\frac{x+n}{2n+1}$$ $$n>1$$ I have this series and I have no idea how to solve it.I have seen the formula but I don't understand it ...
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1answer
38 views

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 $n \leq m$ , $|x_m-x_n|\leq|\frac{1}{n}(\frac{1}{3n+1}+\frac{1}{3n+4}+\cdots+\frac{1}{3m-2})|$ ...
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0answers
44 views

Cauchy sequences according to Cauchy…

(It may be that this question "does not show research effort". Sorry -- it's not a question with mathematical content, so it's impossible to figure out the answer; it's a question about who said what ...
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2answers
50 views

Show that $\{a_n\}$ is also a Cauchy sequence in $X$

Let $\{b_n\}$ be a Cauchy sequence in a metric space $X$ and let $\{a_n\}$ be another sequence in X such that $d(a_n, b_n)<\frac{1}{n}$ for every $n\in \Bbb N$. Show that $\{a_n\}$ is also a Cauchy ...
3
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2answers
111 views

How to show that $x_n=(1+\frac{1}{n})^n$ is a cauchy sequence in $\mathbb Q$. [duplicate]

How to show that $x_n=(1+\frac{1}{n})^n$ is a cauchy sequence in $\mathbb Q$. I am trying to use binomial theorem to expand $(1+\frac{1}{n})^n$ but it is not coming.
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1answer
68 views

A proof related to the convergence of Cauchy sequences

In this problem, we will investigate a new property which some sequences may have. Here is a new definition. Definition: A sequence $\{a_n\}_{n=0}^∞$ is said to be Cauchy iff $$\forall \...
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4answers
38 views

Convergence of Cauchy's sequence

I understood that every convergent sequence is a Cauchy sequence. It seems that the converse is not necessarily true. An example given is the sequence $\{x_n\}$, where $x_n = (0.1)^n$ is a Cauchy ...
12
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2answers
121 views

If a sequence $\{x_{n}\}$ is a Cauchy sequence and the sequence has a limit point $x_{0}$ then $x_{n} \rightarrow x_{0}$

I am trying to prove that if a sequence $\{x_{n}\}$ in a set $M\in X$ ($X$ is a metric space) is a Cauchy sequence and the sequence has a limit point $x_{0}$ then $x_{n} \rightarrow x_{0}$. I wish to ...
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2answers
50 views

Usefulness of Cauchy sequences

I took two courses in single- and multivariable calculus. Both of which dealt with Cauchy sequences. My question is now, why is the property of being a Cauchy sequence useful? I know that it is used ...
1
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1answer
53 views

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.
1
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1answer
37 views

Convergence of sequence from one space with a limit in the other

The usual definitions of a (strong) convergence and a weak convergence of a sequence $(x_n)_{n \geq 1}$ go something like this: Definition 1: A sequence $(x_n)_{n \geq 1}$ in a normed space $X$ ...
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2answers
28 views

The assumption of a Cauchy sequence in proving completeness of a set.

In a lot of proofs in proving that a set is complete we have to go about taking a Cauchy sequence in the set. I've always questioned why are we allowed to "just" do this? As in how are we certain that ...
0
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1answer
66 views

How to calculate $ \lim_{n\to \infty}{\sqrt[n]{a^n+b^n}} $ [duplicate]

I am trying to calculate the following limit of this sequence $ \lim_{n\to \infty}{\sqrt[n]{a^n+b^n}} $. where $ a,b \in \Re $ When nth roots are involved in calculating limits of sequences, we ...
2
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2answers
37 views

Problem understanding a proof for the existence of limit of a real Cauchy sequence.

Let {$x_n$} be a Cauchy sequence of real numbers. The proof I am reading uses the completeness axiom to prove the hypothesis. The proof starts with a lemma stating that this sequence must be bounded, ...
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0answers
24 views

Completion of Metric Spaces

I am trying to show that if $(M,d)$ is a metric space, then if $C$ is the set of all Cauchy sequences on M and $\sim$ is an equivalence relation on $C$ given by $(x_n) \sim (y_n)$ iff $\lim d(x_n, y_n)...
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0answers
102 views

Prove: $\lim\limits _{n\rightarrow\infty} \left( ((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}} \right)=\frac{1}{e}$ [duplicate]

I see this solution in a book but I don't understand it. Which theorems were used? I know how I solve this limit but this solution in book as following: $$\lim\limits_{n \rightarrow \infty} \left(...
0
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2answers
67 views

Show that $\ell^1$ is non complete under $\ell^2$ norm

I'm trying to show that the metric space $ \ell^1$ is not complete under $\left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ So I'm trying find a non convergent cauchy sequence in $\ell^1$ w.r.t $\...
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1answer
26 views

Measure of Cauchy sequence

I consider $(f_n)_{n\in \mathbb N} \in L^p(\mu)$ and $\vert|f_n-f\vert|_p \rightarrow 0 $ for $ n \rightarrow \infty$ So I can conclude: Let $ \epsilon_k =2^{-k}$ $ \forall k \ \exists n_k: \mu(\{|...
6
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3answers
66 views

Does convergence for Cauchy sequence fail only when the limit is not in the domain?

I am trying to understand how important is the distinction between Cauchy sequences and convergent sequences in normed vector spaces $E$. So far I have only come across examples where the Cauchy ...
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0answers
14 views

Non-convergent Cauchy sequence in $(\mathbb{Z}, \vert \cdot \vert_p)$ [duplicate]

As stated in this answer, $\mathbb{Z}$ equiped with the $p$-adic norm $\vert\cdot\vert_p$ is a metric space with the completion $\mathbb{Z}_p$ being the ring of $p$-adic integers. Of course this means ...
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2answers
41 views

C[a,b] with Sup-Norm is complete

I came across many pages claiming to prove that $C[a,b]$ with the Supremum norm: $||f(t)|| = sup_{t\in[a,b]}|f(t)|$ is complete, i.e. that every Cauchy-sequence with elements in $C[a,b]$ converges to ...
0
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1answer
29 views

Proving that a sequence in $\ell^2$ is a Cauchy sequence

Let $(x^{(n)})_{n\in\mathbb{N}}, x^{(n)}:= \sum\limits_{i=1}^n \frac{1}{i} e_i$, where $e_i$ is the sequence that is $0$ everywhere but $1$ in the $i^{th}$ element. I would like to prove that this ...
0
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1answer
29 views

Non-Cauchy sequence in $A \times B$

Let $A$ and $B$ be closed subsets of a Banach space $X$. Let $\{(x_n,y_n)\}$ be a sequence in $A \times B$. It is known that if $\{x_n\}$ is not Cauchy then there exists $\epsilon_1>0$ such that ...
2
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0answers
32 views

Cauchy Sequence in $C[0,1]$ endowed with $L^2$ norm and $L^\infty$ norm

Take $V=C[0,1]$ with the usual $L^2$ norm, i.e., $\|f\|_2^2 = \int_{0}^{1}|f(\tau)|^2d\tau$. Is it complete? Consider the following sequence of functions: $$f_n(t) = \begin{cases} 0 & t &...
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2answers
23 views

Questioning dense subset completeness (counterexample)

Let $X$ be a separable metric space and $A \subset$ X be countable and dense. Characterize the statements below as true or false (and why). If every Cauchy sequence in $A$ converges in $X$, $...
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1answer
38 views

Cauchy sequence confusion

Cauchy sequence definition: "$\forall \epsilon>0, \exists N$ such that $\forall n,m>N, |a_{n}-a_{m}|<\epsilon$". I was told that it is not sufficient to consider $m=n+1$ but however, I ...
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0answers
25 views

Why is it necessary to first reduce our case to a finite $J_{2} \in \mathbb N$ to show completeness of $\ell^{\infty}$

Let $(x^{(n)})_{n}\subset\ell^{\infty}$ be a Cauchy sequence (w.r.t. $\vert \vert \cdot \vert \vert _{\infty}$). Thus for any $\epsilon >0 $ there exists $N \in \mathbb N$ so that for all $n,m \geq ...
0
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1answer
13 views

Convergence within a metric space with different starting points

I'm reading through Terence Tao's Real Analysis II, and he made a seemingly off-hand comment that made me pause and think. "If $(x^{(n)})_{n=m}^\infty$ converges to $x$, then $(x^{(n)})_{n=m'}^\infty$...
3
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3answers
50 views

Given a Cauchy sequence $a_n$, show that $\sqrt{a_n}$ is Cauchy when $a_n>0$ for all $n$.

We have a sequence $a_n$, that is Cauchy and every term is positive. How do I find that $\sqrt{a_n}$ is also Cauchy? I have seen a similar question posted but in that question $a_n>1$ so it is not ...
4
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1answer
70 views

Discover whether $\sum_{n = 2}^{\infty} \frac{1}{n\log(n)}$ is convergent or not, using Cauchy

I was asked if $\sum\limits_{n=2}^{\infty} \frac{1}{n\log(n)}$ was convergent or not. I already solved this problem using the integral property, but I wanted to use Cauchy instead. I defined $m,n \...
0
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2answers
33 views

If a sequence is divergent in $\mathbb{R}$ , then it isn't a Cauchy sequence

I need help, I don't know if the following statement is true or false: "If a sequence $\{x_n\}_{n\in\mathbb{N}} \subset \mathbb{R}$ is divergent, then $\{x_n\}_{n\in\mathbb{N}}$ isn't a Cauchy ...
1
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1answer
20 views

Let $(L_n)$ be a sequence defined by $(1/(n+1)) + (1/(n+1)) + … 1/(2n))$. Show that $(L_n)$ is monotonic increasing.

I am having difficulty understanding the sequence in the following question (Exercise 2.26 out of Real Analysis by Howie). Context question: Let $(L_n)$ be a sequence defined by $$\frac{1}{n+1} + \...
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0answers
30 views

functions convergence

$f$ is bounded function defined around $x_0$. For every monotonic sequence $x_n \rightarrow x_0$, $f(x_n)$ is convergent. prove/disprove : all $f(x_n)$ sequences converge to the same limit when $x_n\...
3
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3answers
74 views

How to prove that $\left\{\frac{1}{n^{2}}\right\}$ is Cauchy sequence

How can I prove that $\left\{\frac{1}{n^{2}}\right\}$ is a Cauchy sequence? A sequence of real numbers $\left\{x_{n}\right\}$ is said to be Cauchy, if for every $\varepsilon>0$, there exists a ...
0
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4answers
40 views

If $(x_n)_{n = 1}^{\infty}$ is Cauchy, show subsequence $(x_{n_{k}})$ such that $\sum_{k = 1}^{\infty}|x_{n_{k}} - x_{n_{k+1}}| < \infty$

If $(x_n)_{n = 1}^{\infty}$ is Cauchy, show that it has a subsequence $(x_{n_{k}})$ such that $\sum_{k = 1}^{\infty}|x_{n_{k}} - x_{n_{k+1}}| < \infty$ Attempt: Since $x_n$ is Cauchy and since $\...
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1answer
29 views

Dedekind cut corresponding to the limit of a Cauchy sequence

Let $a : \mathbb{N} \rightarrow \mathbb{Q}$ be a Cauchy sequence of rationals. Then is it correct to say that $$\lim_{n \rightarrow \infty} a_n = \{x \in \mathbb{Q} : \exists n \in \mathbb{N} : \...
2
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1answer
26 views

Proving this sequence converges in $L^2(\mathbb{P})$

We have some IID sequence, $\left\{ {{X_n}} \right\}_{n = 1}^\infty $, of standard normal random variable on the probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$. Also $\left\{ {{\...
1
vote
1answer
52 views

Show that $(X, \vert\vert\vert \cdot \vert \vert \vert)$ is a Banach Space and that $(X, \vert \vert \cdot\vert \vert_{1})$ is not)

Let $X:=\{ x \in \ell^{1}:\vert\vert\vert x \vert \vert \vert< \infty\}$, and that $\vert\vert\vert x \vert \vert \vert=\sum\limits_{j=1}^{\infty}j\vert x_{j}\vert$ $a.$ Show that $(X, \vert\vert\...
3
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3answers
108 views

Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
0
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1answer
35 views

Convergence of a sequence… [duplicate]

Let $\{a_n\}$ be a sequence of real numbers. Define $\sigma_n = 1/n(a_1 + \dots + a_n)$. Suppose that $\lim a_n = a \in \mathbb{R}$. Show that $\lim \sigma_n = a$. Here is my work so far... Fix $\...
0
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3answers
23 views

Proving something is not a Cauchy sequence (Theory Proof)

I need to prove that ${\{X_n\}}$ is not a Cauchy sequence. I understand that in order to prove this, I need to prove that $$$$$\exists\ \epsilon\gt0\ | \forall N \in \Bbb N,$ if, there is $n,m \geq N$,...