Questions tagged [cauchy-sequences]

Question 1

In John B. Conway's "A Course in Functional Analysis" on pg 46. #6 says "Show that if $T: H \rightarrow K$ is a compact operator and $\{e_n\}$ is any orthonormal sequence in H, then $||...

Question 2

I'm trying to do a Grafakos exercise that asks me to show that the space $BMO(\mathbb{R}^n)$ is complete. To do this, it suggests using the following approach: Let $f\in BMO(\mathbb{R}^n)$. For any $\...

Question 3

I want to know whether it is true that if a real sequence $\{x_n\}_{n=1}^\infty$ satisfies $\lim\limits_{n\to\infty} n|x_n-x_{n+1}|=0$ then it converges. I guess it is false but I can't find a ...

Question 4

On $BV[0,1]$ we can introduce the following norm: $$\lVert f \rVert_{BV} := \lVert f \rVert_{\infty} + V(f)$$ [Here, $V$ is the total variation.] I want to show that this normed space is a Banach ...

Question 5

TL;DR: Are there any notable textbooks that use the definition $$ \forall\varepsilon>0.\;\exists n.\;\forall r\geq n.\;d(u_r,u_n)<\varepsilon \tag{$\dagger$} $$ rather the more standard $$ \...

Question 6

I am currently revising my Topology notes, and today I have started looking into compact metric spaces. I am trying to prove explicitly the following: Theorem - Let $\left( X, d\right)$ be a compact ...

Question 7

Definition - Consider the set $X=\left\{ 0 ,1 \right\}^{\mathbb{N}}$ and define a function $\mu :X\times X \to \mathbb{N}$ by $$\mu \left(\mathbf{x}, \mathbf{y} \right) = \min \left\{n\in \mathbb{N} \...

Question 8

Problems in Real Analysis Ex 1.6.48 Let $(a_n)_{n \geq 1}$ be a sequence of positive real numbers such that for every $n \geq 1$ $$a_{n+1} \leq a_n + \frac{1}{(n+1)^2}.$$ Prove that the sequence $(...

Question 9

I am trying to learn about Hilbert spaces and I know that a complete inner product product space is hilbert space, I know their definition (Metric space, Vector space, Normed space, Banach space, ...

Question 10

Theorem 2.36 of Baby Rudin states the following. "If $\{K_\alpha\}$ is a collection of compact subsets of a metric space $X$ such that the intersection of every finite sub-collection of $\{K_\...

Question 11

The proposition is written as Let $X$ be a subset of $\mathbb{R}$, and let $f \colon X \to \mathbb{R}$ be a uniformly continuous function. Let $(x_n)_{n=0}^\infty$ be a Cauchy sequence consisting ...

Question 12

The following statement is found in an article of Cauchy sequence on ncatlab. https://ncatlab.org/nlab/show/Cauchy+sequence#definitions More generally, let $S$ and $T$ be sets and let $R ( a , b , t )...

Question 13

I need help proving that the series $\sum_{k = 1}^\infty \frac{|\sin k|}{k}$ diverges, specifically by using Cauchy's convergence test. What I tried, in short, is finding a segment such that the value ...

Question 14

Is there any Banach space other than $l^p$ satisfies the following conditions? There exists a biorthogonal system $(x_i, f_i)_{i \in \mathbb{N}},$ where $(x_i)_{i \in \mathbb{N}} \subset X$ and $(f_i)...

Question 15

Ok, so I was reading the book Banach Algebra Techniques in Operator Theory by Douglas and I was thinking about something. 1.7 Proposition In a Banach space each Cauchy net is convergent. Proof Let $\{...

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Questions tagged [cauchy-sequences]

Compact operator condition

How to prove that Cauchy in BMO then is Cauchy in $L^1$

True or false? If $\lim n|x_n-x_{n+1}|=0$ then $\{x_n\}$ converges. [duplicate]

$BV[0,1]$ with the norm $\lVert \cdot \rVert_{BV}$ is complete.

Textbooks using this variation on the definition of a Cauchy sequence?

Every compact metric space is complete - without any a priory knowledge of compactness [duplicate]

Proving a certain Cantor cube is a complete metric space (by definition) - proof verification

for $a_n >0$ $\forall$ $n \geq 1$, Does the condition $a_{n+1} \leq a_n + \frac{1}{(n+1)^2}$ imply convergence of $(a_n)$? [duplicate]

What is "Completeness of Inner Product Space" and "Cauchy Sequence" and How to Prove it.

Proving that a sequence of nested compact sets whose diameter goes to 0 contains exactly one point

A question about Proposition 9.9.12 in Tao's Analysis 1 (Uniformly continuous functions sends Cauchy sequences to Cauchy sequences)

General Cauchy sequence expression [closed]

How do I prove that the series $\sum_{k = 1}^\infty \frac{|\sin k|}{k}$ diverges? [duplicate]

Existence of a Banach space other than $l^p$ satisfying the following conditions

Question about Cauchy net convergence

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