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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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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)...
3
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3answers
75 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 ...
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1answer
30 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 $\...
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3answers
50 views

Limit of a sequence as $n\to\infty$ [closed]

Let $x_{0}$ be a positive real number and $n\in\mathbb{N}$. Then what is $$ \lim_{n\to\infty}\{(x_0+n)^r-n^r\} $$ where $r\in (0,1)$ is fixed number.
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3answers
22 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 $n,m \geq N$,...
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2answers
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Prove that if a sequence is Cauchy then there exists a sub sequence such that $|a_{n_{k+1}} - a_{n_k}| < 1/k^2$ [closed]

I have to show that if a real sequence $\{a_n\}_{n\in\mathbb{N}}$ is Cauchy then there has to be a sub sequence $\{a_{n_k}\}_{k\in\mathbb{N}}$ satisfying the following: $$\left|a_{n_{k+1}}-a_{n_{k}}\...
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1answer
19 views

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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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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0answers
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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
36 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
65 views

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
29 views

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? ...
0
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1answer
29 views

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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0answers
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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
40 views

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
100 views

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
43 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 ...
0
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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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2answers
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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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1answer
38 views

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\}...
2
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1answer
32 views

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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3answers
43 views

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
36 views

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
27 views

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
30 views

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
65 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
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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
48 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
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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 ...
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2answers
45 views

Some limits of sequences

Prove by definition the following limit: $$\lim_{n\rightarrow \infty}\sqrt{\frac{2n^3+3n-1}{8n^3+n^2}}=\frac{1}{2}$$ For the first i've been tried rationalize: $$\begin{align}\left|\frac{\sqrt{2n^3+...
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1answer
56 views

Proving a sequence is Cauchy in metric

Consider the sequence, $f_n(x)= \begin{cases} (2x)^n & 0 \leq x\leq \frac{1}{2} \\ 1 & \frac{1}{2} \leq x \leq 1\\ \end{cases}$ Then we need to show that $\{ f_n\}$ is Cauchy ...
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2answers
42 views

Prove that if $n\ge 2$ is an integer, then $1/n^2 < 1/(n-1) - 1/n$ [closed]

Prove that if $n\ge 2$ is an integer, then $1/n^2 < 1/(n-1) - 1/n$ Unsure how to start this question! I've tried to prove by induction but I keep getting $n>1$?
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2answers
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How these two definitions are equivalent?

Please have a look at these two equivalent defintions of Cauchy's general principle of convergence series. I understand the first defintion but I'm having problems with the second defintion. First ...
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3answers
40 views

Cauchy sequences of rationals with limit irrational: natural, or geometric examples

As we know, real numbers are constructed by filling up gaps between rationals by the limits of all Cauchy sequences of rationals. Q. What are examples of sequence of rationals such that its easy ...
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1answer
41 views

Suppose $f:[0,1]\times[0,1]\mapsto X$ is a continuous function. Show that $[0,1]\times[0,1]$ can partitioned into rectangles s.t $f(R_i)\subseteq U_k$

Suppose $f:[0,1]\times[0,1]\mapsto X$, is a continuous function where $X$ compact and connected subset of $\mathbb{R}^n$. Show that $[0,1]\times[0,1]$ can partitioned into rectangles $R_i$ such that $...
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1answer
16 views

Completeness of a Normed Space of Smooth, Bounded Functions

As part of a proof of the Picard–Lindelöf theorem, I am using the following space: $X = \{ u \in C([0,T]) : u(0) = \alpha , || u - \alpha || \leq K\}$ where $K \in \mathbb{R}_{> 0} , \ \alpha \in ...
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1answer
52 views

Series limit involving Binomial coefficients

Consider the parameters $v_{a}$, $v_{b}$ be such that $0<v_{a}\leq v_{b}$ and $c>0$. I have an equation involving the Binomial distribution that I need to solve with respect to $p(n)$: $\sum_{k=...
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0answers
28 views

Lower bound $\sum\limits_{n=1}^\infty a_n/(a_n -1)$ in terms of $a_1$, where $a_n$ is an infinite sequence under some conditions.

This is a question related to a previous question ($a_n$ is an infinite sequence with $\sum\limits_{n=1}^\infty a_n\leq1$ and $0\leq a_n<1$. Prove that $\sum\limits_{n=1}^\infty a_n/(a_n-1)$ ...
0
votes
1answer
27 views

$(f(x))_{n\in\mathbb{N}}$ and $(f(y))_{n\in\mathbb{N}}$ have the same limit.

Assume that $f: \mathbb{R} -\{0\}\to \mathbb{R}$ is uniformly continuous. Assume $(x_n)_{n\in\mathbb{N}}\in(\mathbb{R}-\{0\})^\mathbb{N}$ and $(y_n)_{n\in\mathbb{N}}\in(\mathbb{R}-\{0\})^\mathbb{N}$ ...
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2answers
42 views

Sufficiency in the proof that $L^p(\mu)$ is complete

In the proof that $L^p(\mu)$ is complete for $p\in[1,\infty]$ (as done in Saxe, Theorem 3.21 or in Folland, Theorem 6.6, the latter of which is outlined here) we make use of the following completeness ...
0
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1answer
60 views

Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
1
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2answers
28 views

If $A \in \mathcal{L}(H)$ and $\langle A(u),u \rangle \geq \langle u, u \rangle$, then $A$ is invertible.

Exercise : Let $H$ be a Hilbert space and $A \in \mathcal{L}(H)$ such that : $$\langle A(u),u \rangle \geq \langle u, u \rangle \; \forall u \in H$$ Show that $A$ is invertible. Attempt/...
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3answers
62 views

Show $\{x_n = \sqrt{n}\}$ is not Cauchy sequence

Consider the sequence {$x_n$},$x_n$=$\sqrt{n}$ Show that $\forall \varepsilon > 0, \exists n_0 \in \Bbb N$ s.t. $\forall n \geq n_0$, |$x_{n+1}-x_n$|<$\varepsilon$. This is what I have: Let $\...