Questions tagged [cauchy-sequences]
For questions relating to the properties of Cauchy sequences.
2,395
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prove that the sequence $g_n(x)$ is uniformly Cauchy
Let $g : \mathbb{R} \rightarrow \mathbb{R}$ be uniformly continuous and for each $n \in \mathbb{N}$ let
$g_{n}:\mathbb{R} \rightarrow \mathbb{R}$, $x \mapsto g_{n}(x) = g \left( x + \frac{1}{n}\right)$...
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Is this a valid way of proving this series converges?
Take the series $\sum_{k=1}^\infty \frac{(-1)^k}{2^k+k}$
Since I use the Cauchy Criterion, for $n, m$ ∈ ℕ with $m > n$, define:
$S_{m,n} = \sum_{k=n+1}^m \frac{(-1)^k}{2^k+k}$
Here is my (shortened)...
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Example of a sequence $x_n$ which converges to $0$, $n(x_n-x_{n+m})=O(1)$ but $(x_n-x_{n+k}) \notin \ell^1(\mathbb{N})$
Let $(x_n)_{n \in \mathbb{N}}$ be a sequence of real numbers such that $x_n \to 0$ as $n \to \infty$. I want to know if the following conditions are equivalent or if there is one weaker.
$n(x_n-x_{n+...
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Proof that if $f$ is uniformly continuous then for every Cauchy sequence $(x_n)$ with $a < x_n < b$ $f(x_n)$ is also cauchy.
I need to show that the following statement holds true:
Given $a, b \in \mathbb{R}$, $a < b$, $f: (a, b) \to \mathbb{R}$, $f$ continuous. Show that $f$ uniformly continuous $\Rightarrow$ $\forall$ ...
2
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1
answer
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Is there a closed-form definition for this bounded cauchy sequence?
The sequence is
0,1,1/2,0,1/3,2/3,1,3/4,2/4,1/4,0,1/5,2/5,3/5,4/5,1,5/6,4/6,3/6,2/6,1/6,0,1/7,....
it is from the top answer to this question
I considered some ...
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1
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Constructing an RKHS from a Kernel
I'm reading the book "High Dimensional Statistics" by Martin Wainwright just for fun (also as preparation of my PhD in computer science/Machine Learning). In particular, I'm currently ...
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1
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46
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Is the given sequence cauchy , convergent
Question: Let $\\{a_n\\}$ be a sequence that satisfies
$|a_n| < 2$ ,
$|a_{n+2}-a_{n+1}| \leq \frac{1}{8}|a_{n+1}^2-a_{n}^2|$.
Then
a) $\\{a_n\\}$ is a cauchy sequence
b) $\\{a_n\\}$ is a bounded ...
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Proving that if a Cauchy sequence does not tend to zero, then it is bouned away from zero. [duplicate]
Definition: A sequence $(x_{n})$ is bounded away from zero if there exists some bound $b$ such that $\lvert x_{n}\rvert >b$ for all $n$; in particular $x_{n}\neq 0$ for all $n$.
I want to prove ...
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1
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why $t_n\to 0$. [closed]
Let $(t_n)$ and $(v_n)$ be two sequence such that $\forall n\in\Bbb N: 0\leq t_n\leq 1$ and $v_n\geq 0$ with $v_n\to 0$.
Assume $\forall n\in\Bbb N: 0\leq t_{n+1}\leq \frac{t_n+v_n}{M}$ where $M&...
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1
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the description of Cauchy principle of convergence
Question：Is description
$$\forall\varepsilon\gt0,\exists N\in\mathbb{Z^+},\text{s.t.}\forall n\gt N,\left| a_n-a_N\right|\lt\varepsilon$$
equivalent to convergence of the suquence $\{a_n\}$?
Attempt:
...
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Does the existence of limit of a sequence formed by continuous functions at some points imply the existence of the limit at other points?
$\{A_i,i\in\mathbb N\}$ is a fixed matrix sequence with element $A_i\in \mathbb R^{n\times m}$.
$\Phi\in\mathbb R^{m\times m}$ is a constant matrix and $d\in\mathbb R^m$ is a vector.
The sequence $\{...
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Some wrong when I use Cauchy sequence
Let $x_n = \sqrt n $,
$\forall p \in \mathbb{N} ,\forall ε>0,\exists N=[\frac{p^2}{ε^2}]+1,n>N$,
$|x_{n+p}-x_n|=\frac{p}{\sqrt{n+p}+\sqrt n}<\frac{p}{\sqrt n}<ε$
So the sequence$\{x_n\}$ ...
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Question about Cauchy sequence
By definition of Cauchy sequence, if $(a_n)$ is a Cauchy sequence, then we have $|a_n-a_m| \rightarrow 0$, as $n,m \rightarrow \infty$.
Suppose $(b_n)$ is a sequence. For any $m\in \mathbb{N}$, we ...
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Proof that the sequence is a Cauchy sequence.
Let $a \in (0, 1)$ be a number with decimal representation:
$0.a_1 a_2 a_3 \ldots,$
where $a_k \in \{0,1,\ldots,9\}$ for $k \in \mathbb{N}$. Show that the sequence $(x_n)$ with:
$x_n = \sum_{k=1}^{n} ...
1
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1
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Problem on cauchy sequences
$\left\{\frac{1}{n}\right\}$ is a Cauchy sequence. Determine $N_0$ such that $|u_n - u_m| < 0.021$, whenever $m, n > N_0$.
a. $48$
b. $45$
c. $46$
d. $47$
The correct answer for this was $48$. ...
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1
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Proof that a sequence defines a Cauchy sequence
I recently did a question that said to prove the sequence ${\{x_n\}}=\frac{3n}{2n+1}$ is Cauchy and I'm unsure on if my proof is valid.
First I stated that ${\{x_n\}}$ is Cauchy $\iff \forall \epsilon ...
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How to show that if a rational sequence $(x_n)$ is Cauchy, then $(1/x_n)$ is also Cauchy
I am struggling to prove the following:
If $(x_n)$ is a Cauchy sequence in $\mathbb Q$ with $x_n\nrightarrow0$, then the sequence $(1/x_n)$ is also Cauchy.
My attempt at proof:
Let $\epsilon>0$. ...
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Cauchy conversion Criteria Integral
I have the following question: I want to show, that the improper integral exists:
$$\int_{0}^{b}\tfrac{\sqrt{1+y'(x)^2}}{\sqrt{y(x)}}$$
I stated: For $$[t,b] \subset(0,b]$$, all t must be in the ...
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Is this solution correct for this problem about cauchy sequences?
My first thought is that $x = a^{1/k}$, so of course $x$ is unique since $a^{1/k}$ is a specific fixed value. But this seemed way too simple of a solution so I think I am misunderstanding the problem.
...
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In a complete space $X$ is every $x \in X$ the limit of a sequence $\{x_n\}$ such that $x \not\in \{x_n\}$? [closed]
Let $X$ be a complete metric space. Then for any point $x \in X$, can it be shown that there exists a sequence $\{x_n\} \in X$ such that $x \not\in \{x_n\}$ and $x_n \rightarrow x$? More generally, I ...
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Doubt about continuity of a function
Consider the function $f : \mathbb{R} \longrightarrow \mathbb{R}$ defined by
\begin{equation*}
f(x)=\begin{cases}
1 + \frac{1}{q}, \quad &\text{if} \, x = \dfrac{p}{q} \in \mathbb{Q}. \\
...
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if (un) be a monotone bounded sequence, prove that exactly one of l.u.b and g.l.b. of (un) does not belong to (un)
My attempt
Let's assume that the sequence $(u_n)_n$ is both monotone (either increasing or decreasing) and bounded. We need to prove that exactly one of the least upper bound (l.u.b) and greatest ...
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Show that $C([0, 1])$ equipped with $\Vert f\Vert_1:=\int_0^1|f(x)|dx$ is not complete
Show that the normed space of continuous real valued functions $C[0,1]$ equipped with the norm $\Vert f\Vert_1:=\int_0^1|f(x)|dx$ is not complete.
Let be $f_n:[0,1]\to\mathbb{R}$ with $f_n(x)=x^n$. ...
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What is the difference between this alternate definition of Cauchy and its actual definition? Will this one work?
The definition of Cauchy sequence is:
for any $ε > 0$, there exists a natural number $N$ such that if $m, n ≥ N$, then $|a_m − a_n| < ε$.
What if we changed the definition to:
for any $k ≥ 1$, ...
1
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1
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How to check the convergence or divergence of sequence $\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{\sqrt{3}\sqrt{3}},\dots$?
I tried like this
$a_n=\sqrt{3\sqrt{3}\sqrt{3}\dots}$ $n-times$
Taking square of the both sides
$(a_n)^2=3a_{n-1}$
$\Rightarrow \dfrac{(a_n)^2}{a_{n-1}}=3$
But I don't know how to proceed further.
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Symbolizing Definition of Cauchy Sequence with Predicate Logic
I am trying to symbolize the following definition of a Cauchy Sequence with the syntax of predicate logic. Can someone please take a look and tell me if I am symbolizing the definition correctly ...
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1
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Confusing notation of sequences in $k$-dimensional Euclidean spaces.
Suppose $(x^{(n)})$ is a sequence in $\mathbb R^k$, $k \in \mathbb N$. From what I understand, $(x^{(n)})$ is a sequence of sequences $(x_1^{(n)}, x_2^{(n)}, x_3^{(n)}, ..., x_k^{(n)})$. Or in other ...
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Extracting pointwise convergent subsequence. Is my proof correct?
Let $A\subset \mathbb{R}^{d}$ be some bounded set and $E_{n}\subseteq A\;\forall n\in\mathbb{N}$. Assume $\{E_{n}\}_{n=1}^{\infty}$ is a Cauchy sequence with respect to metric $\rho$, defined as $\rho(...
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If $s_n$ is a sequence that converges, then show that if $s_n \ge a$ for all but finitely many $n$, then $\lim s_n \ge a$
This question has been asked multiple times, but none of them use the same method I have used to attempt the proof.
Here's what I did:
$s_n$ is a convergent sequence. Thus
$$\left|s_n - s\right| \lt \...
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Prove the Cauchy Convergence for Brachystochrone
I used Cauchy Criteria to show that there is an Integral for the solution of the Brachistochrone and got $$\Big| \int_{t_{k}}^{b}f(x)dx - \int_{t_{l}}^{b} f(x)dx\Big| = \Big| \int_{t_{k}}^{t_{l}}f(x)...
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Sol Verif: Prove that if $\lim_{n \rightarrow \infty }\sum u_n=l\Rightarrow \lim_{n \rightarrow \infty} u_n=0$ via Cauchy
Question:
Prove that if $\lim_{n \rightarrow \infty }\sum u_n=l\Rightarrow \lim_{n \rightarrow \infty} u_n=0$ by using Cauchy definition of convergence.
My answer:
1- First of all let's write: $S_n=\...
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Showing that $(XY_n)$ is a Cauchy sequence when $(Y_n)$ is cauchy in space of polynomias.
Letting $\mathbb{C}_0[X]$ be the space of complex polynomials without constant term and $(\kappa_n)$ a sequence of real numbers (To be precise, the $\kappa$'s are the free cumulants of a compactly ...
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Existence of Bound of Width $\epsilon$ for Non-Cauchy Rational Sequence
Let $\{x_n\}$ be a sequence of Rational numbers: $$\exists \ \ 0 < \epsilon \in \Bbb Q \ \ \exists \ \ N \in \Bbb N: \lvert x_n - x_m \rvert < \epsilon \ \ \forall \ \ n,m \geq N$$
This is not ...
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Example of a sequence such that $(n^2(a_n-a_{n+2} -(a_{n+2} -a_{n+4})))$ is bounded but $(n^2(a_n-a_{n+2}))$ is not bounded
Let $(a_n)_{n \in \mathbb{Z} }$ be a real sequence which converges to $0$ as $|n| \to \infty$. It can be shown that if the sequence $(n^2(a_n-a_{n+2}))_n$ is bounded, then the sequence
$$(n^2(a_n-a_{n+...
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Monotone convergence theorem - what does non-decreasing exactly mean?
I understand that if a sequence is bounded by supremum and is strictly increasing it will converge. It is intuitive because the sequence is strictly increasing.
I do not really understand the weaker ...
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Prove that the limit of the sequences corresponding to the image of two sequences converging to the same point is the same.
Context : I consider $(E, d_E)$ and $(F, d_f)$ two metric spaces and $A\subset E$ a dense subset of $E$ (i.e $\bar{A}=E$). The function $f$ is defined only on $A$.
I would like to prove that if I have ...
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Why do we use this norm on sequence spaces
I'm studying sequence spaces $\ell^p=\{(x_j)_{j\in \mathbf{N}}:\sum_{j\in \mathbf{N}}|x_j|^p<\infty\}$ for $1\leq p<\infty$.
This is a vector space (I'm not sure how to prove it is closed under ...
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0
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Is a Cauchy sequence in R necessarily a Cauchy sequence in Q
Let $(x_n)$ be a Cauchy sequence in the metric space $\mathbb{R}$ with the Euclidean metric, with the property that $x_n\in\mathbb{Q}$ for all $n\in\mathbb{N}$.
Is it true that $(x_n)$ in the metric ...
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70
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Cauchy Sequence Definition and Theorem
Both these theorems were discussed in class, I wanted to discuss some doubts regarding theorem 1. Is it possible to say simply from theorem 1 alone that any sequence for which $\left|x_{n+1}-x_n\right|...
1
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2
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61
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Cauchy Sequence Contractive Condition
Show that the sequence $\left(x_n\right)$ satisfies the cauchy criterion
$x_1=1$ and $x_{n+1}=\frac{1}{2+x_n^2}\:\forall \:n\ge 1$
As per the hint for this problem, they just follow the regular route ...
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If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^0 (Y)$, then $(f_n)$ is Cauchy in $L^0 (Z)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of ...
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A property of complete metric spaces makes them length (path or inner) metric spaces, Clarification of a proof
In the book "Metric Structures for Riemannian and Non-Riemannian Spaces", by Misha Gromov, I found a proof of the following statement (of Theorem 1.8. restated here more concentrated)
Let $(...
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35
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If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$, then $(f_n)$ is Cauchy in $(L^0 (Z), \rho_Z)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of ...
- 5,427
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2
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Example of a sequence that is Cauchy in a stronger norm and convergent in a weaker norm, but not convergent in the stronger norm?
A norm $\|\cdot\|_1$ on a normed vector space is called stronger than $\|\cdot\|_2$ when $\|x\|_2\leq M\|x\|_1$ for some $M>0$ and all $x$. It is a standard trick (e.g. in proving completeness) to ...
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Are Cauchy sequences for Hilbert space an expression of a compact (multiplication) operator?
Background: I'm reading about Hilbert spaces that require a complete metric space using inner product, where every Cauchy sequence of points $x_m$,$x_n$ on the metric space, $X$, has a limit, also in $...
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2
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Convergence of a "shuffled" sequence
Let $ \{\alpha_n \}_{n \in \mathbb{N}} \subseteq (0, 1) $ be a sequence.
From it, define the following sequence $\{\tau_n\}_{n \in \mathbb{N}}$ by:
$$\tau_0 = 1, \tau_1 = 0, \tau_{n+2} = \alpha_n\...
1
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1
answer
77
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Proving that every cauchy sequence is bounded: Why is the set $\cal N$ internal?
I'm having some trouble understanding the following proof that every real Cauchy sequence is bounded:
Proof:
Let $(a_n)_{n\in \mathbb N}$ be a Cauchy sequence and suppose, for the sake of ...
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2
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2
answers
173
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Power series expansion for sine function
I wonder if the power series expansion of sine is a Cauchy sequence, i.e. the sequence $(\sum_{i=0}^{j}\frac{(-1)^{i}}{(2i+1)!}r^{2i+1})_{i\in\mathbb{N}}$ is Cauchy? Let $\varepsilon>0$. Is it ...
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1
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47
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Convergence of a sequence in $L^2$
Let $\{a_k\}_{k\ge 0}$ a bounded sequence in $L^2(\Omega)$ and $a \in L^2(\Omega)$. I can prove the following inequality:
$\left\lVert a_k - a \right\rVert_{L^2(\Omega)}^2 \le \left\lVert a_{k-1} - ...
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1
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Alternative proof of 3.11(b) in Baby Rudin?
I'm self studying Principles of Mathematical Analysis by Walter Rudin (so I welcome any feedback), and I'm wondering if there is any fault to this self written proof, which I think is simpler than the ...
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