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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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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 ...
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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 ...
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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
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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
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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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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 ...
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1answer
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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 $(x^{(n)})_{n=m'}^\infty$...
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3answers
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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 ...
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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 \...
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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 ...
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1answer
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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
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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\...
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3answers
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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 ...
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4answers
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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
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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} : \...
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1answer
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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\{ {{\...
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1answer
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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\...
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Problem Concerning Cauchy Principle for Sequences.

I have a question but can't seem to figure out how to solve it. The problem states: Let's consider a sequence $x_n$, such that $x_n\to a$, as $n \to \infty$. Using the Cauchy Principle prove that (a)...
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3answers
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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 ...
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1answer
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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 $\...
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3answers
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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 $n,m \geq N$,...
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1answer
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Uniform Cauchyness

Assume the sum of $|f_n|$ is uniformly cauchy. Does this imply that the sum of $(f_n)$ is uniformly cauchy? My reasoning is yes, since every $f_n$ is bounded by $|f_n|$.
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2answers
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Cauchy sequence with $\{ x_n : n \in \mathbb{N} \}$ not closed converges

Suppose $(x_n)_{n \in \mathbb{N}}$ is a Cauchy sequence and $A = \{ x_n : n \in \mathbb{N} \}$ not closed. Show that there exists $x \in X$ such that $x_n \longrightarrow x$. Since $A$ is not closed:...
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0answers
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Why is $\sup_{t \in A} |x_m(t)-x(t)| \leq \sup_{t \in A} \lim_{k \rightarrow \infty} |x_m(t)-x_k(t)|$?

Why is $\sup_{t \in A} |x_m(t)-x(t)| \leq \sup_{t \in A} \lim_{k \rightarrow \infty} |x_m(t)-x_k(t)|$? $(x_n)$ is Cauchy, it's in $BoundedLinear(A,\mathbb{K})$, $t \in A$. I'm thinking that this ...
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On the convergence of a sequence of functions

Suppose that $f_n → f$uniformly on some set $E$ and that for each $n$, there exists $M_n$ such that $$|f_n(x)| ≤ M\quad\text{for all }n=1,2,3...\text{ and all }x ∈ E.$$ Suppose $g$ is a continuous ...
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1answer
38 views

If ${x_{n}}$ is a sequence of real numbers satisfying a certain condition, then which of the following is true

Suppose that $\{x_{n}\}$ is a sequence of real numbers satisfying the following. For every $\epsilon > 0$, there exists positive integer $n_{0}$ such that $\left |x_{n+1}-x_{n}\right |$<$\...
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0answers
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Prove the sequence $f_{n} = \frac{1}{n^2+1}$ is a Cauchy sequence.

Prove the sequence $f_{n} = \frac{1}{n^2+1}$ is a cauchy sequence. I'm just making sure my logic and reasoning is sound for the above proof: Definition of cauchy sequence: $f_n$ is Cauchy if for all ...
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2answers
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Define a sequence $\{y_n\}$ by $y_{2n-1}=x_n$ and $y_{2n}=x$. Which of the following are true?

Let $(X,d)$ be a metric space and let $x_n$ be a sequence in $X$. Let $x\in X$. Define a sequence $\{y_n\}$ by $$y_{2n-1}=x_n,y_{2n}=x,n\in \mathbb N$$ Which of the following statements are correct? ...
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1answer
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Can we say that if every convergent sequence in X is Cauchy then X is a Banach space?

I know that if X is Normed linear space then every convergent sequence in X is Cauchy.. is it true the other way around or does it all actually mean one thing ??
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Will a convergent Double Sequence be bounded also?

A convergent Double Sequence will be bounded also. My Attempt: I think the statement is not true. Counter Example : $a_{1n} = n$, $ a_{mn} = 1/m + 1/n$ for all $m \geq 2$ lim$_{m,n ...
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1answer
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Show that a sequence is a Cauchy sequence

Suppose that $(x_n)$ is a decreasing sequence of non-negative real numbers that converges to 0. Prove, using the definition, that the sequence $(y_n)$ where $y_n = x_1 - x_2 + x_3 - x_4 + \cdots + (-...
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2answers
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Prove the alternating sum of a decreasing sequence converging to $0$ is Cauchy.

Let $(x_n)$ be a decreasing sequence with $x_n > 0$ for all $n \in \mathbb{N}$, and $(x_n) \to 0$. Let $(y_n)$ be defined for all $n \in \mathbb{N}$ by $$y_n = x_0 - x_1 + x_2 - \cdots + (-1)^n x_n ...
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1answer
45 views

Interpreting meaning of “Cauchy series” (not sequence)

I am trying to prove that some particular infinite series is Cauchy. Of course, I know what it means for a sequence to be Cauchy, but I can't seem to find anything related to Cauchy series. Could ...
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1answer
31 views

$\ell^p(J)$ is complete (Banach)

I have to prove that the space $\ell^p(J)$ defined as the set of all functions $\psi: J\rightarrow \mathbb{F}$ s.t. $\psi$ is null except in a contable subset of $J$ and $||\psi||_p :=\bigg(\...
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Convergence of power series beyond radius of convergence?

In rudin analysis text book, 3.44 theorem , it says about the convergence on the boundary of the circle of the power series, the theorem is roughly:- Suppose the radius of convergence of $\Sigma ...
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Let $g_n(x)=\sum_{k=1}^n (-1)^k f_k(x) \forall x\in \mathbb R. $ Then which one of the following are correct answers?

Suppose that $\{f_n\}$ is a sequence of continuous real-valued functions on $[0,1]$ satisfying the following: (A)$\forall x\in \mathbb R,\{f_n(x)\}$ is a decreasing sequence. (B)the sequence $\{f_n\}...
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1answer
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Slightly alternative proof to the converse part of Cauchy's General Principle

I want to prove that: If $\forall \epsilon >0$, $\exists k \in \mathbb{N}$, such that $| u_{n+p}-u_n| <\epsilon $, whenever $n\geq k$, $p\in \mathbb{N}$, then $\{u_n\}$ is convergent. Proof: [...
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Intuitively, $\sqrt{n}$ is not convergent. However, $|\sqrt{n+p}-\sqrt{n}|<\epsilon, \forall \epsilon>0, p\geq 1$

$|\sqrt{n+p}-\sqrt{n}|<\epsilon$ Clearly, $|\sqrt{n+p}-\sqrt{n}|=\frac{p}{\sqrt{n+p}+\sqrt{n}} \leq p/\sqrt{n} \rightarrow 0$. But by definition of a cauchy sequence, if we can choose $\exists N: ...
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1answer
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Determine if Sequence is Cauchy

Can someone please tell me how to determine if a sequence is Cauchy without using the limit. I'm supposed to use partial fraction decomposition $a_n=\frac{1}{n(n+1)}$. When I did the partial ...
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1answer
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The convergence of a bounded sequence ${x_n}$ satisfying $x_{n+1} - \epsilon_n \le x_n$, where $\sum_{n=1}^\infty \epsilon_n$ is absolutely convergent

Statement: If a bounded sequence $\{x_n\}_{n=0}^\infty$ in $\mathbb{R}$ satisfies $x_{n+1} - \epsilon_n \le x_n$ for $n \in \mathbb{N}$, where $\sum_{n=1}^\infty \epsilon_n$ is an absolute convergent ...
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2answers
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Cauchy sequence in a discrete space

how to prove that any Cauchy sequence in a discrete space is stationary Let $(x_n)$ be a cauchy sequence then $$\forall \varepsilon>0, \exists n_0\in \mathbb{N},\forall p,q \geq n_0\Rightarrow \...
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0answers
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Notation in Cauchy Sequence and Convergence

I have been reading lecture script and wasn't sure where this inequation for a fix-point iteration comes from and what it means. For $k, j \geq 0$ $$ \left|x_{k+j}-x_{k}\right| \leq \sum_{i=0}^{j-1}\...
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1answer
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Proving that $l_\infty$ is complete

I'm learning about Hilbert spaces and operators theory, from some book. I came across the following question - And the books' answer: What I don't understand in the proof - Why can we understand ...
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1answer
66 views

$|x_{n + 1} - x_n| < \frac{1}{2^n} \Rightarrow (x_n)$ is Cauchy [duplicate]

Let $(x_n)$ be a real sequence with the property that for all $n \in \mathbb{N}$, $$|x_{n + 1} - x_n| < \frac{1}{2^n}$$ I want to show, using the definition of a Cauchy sequence, that $(x_n)$ must ...
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2answers
46 views

$n$ tending to infinity

What does n tends to $\mathbf\infty$ mean ? Is it equivalent to saying $n>K$, $\;K\in \Bbb N$ ? For a Cauchy sequence, is $|a_m-a_n|<\varepsilon, \enspace m,n >K$ equivalent to saying $a_m-...
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0answers
35 views

Direct proof for bounded monotonic sequence is Cauchy

Show that every bounded and monotonic sequence is Cauchy. My proof: Suppose $(a_n)$ is a bounded and monotonic sequence. Since $(a_n)$ is bounded, it follows that $\exists M \in \mathbb{Q}^{+}$ ...
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1answer
51 views

Convergence of $\sum_{j=1}^\infty x_j^2$ assuming that the sum of squares is finite.

If I let $l_2$ be the set of all real sequences $\{x_j\}_{j\in N}$, such that $\sum_{j=1}^{\infty} x_j^2 < \infty$, is there any way to show that this sum converges? Can I do it by showing that $\...
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3answers
50 views

Is the following sequence $x_{n}=\frac{1 +(-1)^n}{n}$ Cauchy?

Is the following sequence $x_{n}=\frac{1 +(-1)^n}{n}$ Cauchy? I got not Cauchy, but would appreciate someone to check this. Thank you! So I got the sequence is bounded by 2 and that $abs(x_n-x_{n+1})&...
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1answer
28 views

Proving a sequence is Cauchy from uniform continuity

Given: $f: X \longrightarrow Y$ is uniformly continuous on $X$, $(x_n)_n \in X $ is a Cauchy sequence. Question: What can you say about the sequence ${f(x_n)}$ ? My attempt: Since $f$ is uniformly ...
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4answers
2k views

Can a Cauchy sequence converge for one metric while not converging for another?

Is there an easy example of one and the same space $X$ with two different metrics $d$ and $e$ such that one and the same sequence $\{x_n\}$ is a Cauchy sequence for both metrics, but converges only ...