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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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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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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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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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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 $\sum\limits_{n=1}^\infty a_n/(a_n-1)$ ...
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$(f(x))_{n\in\mathbb{N}}$ 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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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 ...
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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 ...
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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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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 $\...
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Let $x_n$= $\sum_{k=1}^{n}$ 1/k! = 1+1/2!+1/3!+…+1/n! show that the sequence {$x_n$} is Cauchy.

Let $x_n = \sum_{k=1}^{n} 1/k! = 1+1/2!+1/3!+...+1/n!$ for each $n \geq 0$. Show that the sequence $\left(x_n\right)$ is Cauchy. This is what I have: for $n>m$, \begin{align} &|x_n-x_m| \\ &...
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Proving that the metric space of all sequences of positive integers is complete

Consider the set of all sequences of positive integers with the following metric:given $x=(n_j)$, $y=(m_j)$ $$d(x,y)= 1/\inf\{j: n_j \ne m_j\}$$ if $x\ne y$ and $0$ otherwise. I want to show that it ...
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Complete metric space on unique metric

Consider the metric $$d(m,n) = \frac{|m-n|}{mn}$$ Is this metric on the natural numbers $(1,2,\ldots)$ complete? I'm struggling but heres an idea I have from reading other similar questions. The ...
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Proving that a sequence converges if $|a_n - a_{n+1}| < Mr^n$ for some $M > 0$ and $r \in (0,1).$

Let $\{a_n\}_n$ be a sequence. Suppose that there exist $M \gt 0$ and $r \in (0, 1)$ such that $|a_n - a_{n+1}| \lt Mr^n$ for all $ n \in \Bbb N.$ Prove that $\{a_n\}_n$ converges. I'm not really sure ...
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Verification of proof of nonconvergence

I'm not sure if my proof is sound or not, I was wondering if anyone could verify. Prove that the sequence $\{a_n\}_n $ given by $a_n = \frac{(-1)^n\;n+1}{3n}$ does not converge. Proof: Let $\epsilon ...
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Let there be a sequence such that the distance between two consecutive terms converges to 0. Must this sequence converge? [duplicate]

I'm trying to solve the following analysis problem and I've developed a proof, I'm just not entirely sure if it's valid or not. Let $\{a_n\}_n$ be a sequence such that $|a_n -a_{n+1}|\to 0$. Must $\{...
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Proof of X is not Banach

Set $X = \lbrace u\in\mathcal{C}^2 [0,\pi]: u(0)=u(\pi)=0\rbrace$ equipped with the norm $$\Vert u \Vert = \left(\int_0^\pi (u'(x))^2\ dx + \int_0^\pi u(x)^2\ dx\right)^{1/2}$$ I want to prove that $(...
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Does $\lim\limits_{x \rightarrow c} f(x)$ exist if the sequence $\{ f(x_n)\}_{n=1}^\infty$ is Cauchy?

I'm struggling a little with this question: Let c be a cluster point of $A ⊂ \mathbb{R}$, and $f : A → \mathbb{R}$ be a function. Suppose for every sequence $\{x_n \}$ in A, such that $\lim x_n = c$,...
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False theorem on particular case of Cauchy Sequences — why is it wrong?

It goes as follows: If $|x_{n+1} - x_n| < \epsilon$, $\{x_n\}$ is Cauchy. We have: $$\forall \epsilon_0 > 0, \exists N : n > N \implies |x_{n+1} - x_n| < \epsilon_0$$ Let $m,n \in \mathbb{...
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Question on operator norm $\vert \vert \cdot \vert \vert$

I have just been introduced to the operator norm $\vert \vert \cdot \vert \vert$ on $L(X,Y)$ where $X, Y$ are respective normed spaces and any $T \in L(X,Y)$ is simply a linear map from $X$ to $Y$. ...
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Proof of divergence for complex number non-cauchy sequence

So, I trying to prove the following: "Show that a complex number sequence converges, iff, it is a Cauchy sequence". Let $\{w_n\}$ be a complex number sequence. For every $\epsilon>0$,$\exists N=N(\...
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Is this normed vector space complete?

The normed vector space in question is $(C^{1}[0,1],\lvert \lvert \cdot \rvert \rvert _c )$ where $\lvert \lvert f \rvert \rvert _c := \lvert f(1) \rvert + \lvert \lvert f' \rvert \rvert _{1}$ For ...
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Suppose V is a linear subspace of $\mathbb{R}^n$. How to show it's closed in $\mathbb{R}^n$

We want to show that if $V\subseteq\mathbb{R}^n$ is a linear subspace of $\mathbb{R}^n$, then it's closed. Let $V\subseteq\mathbb{R}^n$ arbitrary e suppose $V$ linear subspace of $\mathbb{R}^n$. As ...
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Multiplying convergent sequence by divergent sequences of natural numbers

Let $\{x_k\}$ be a sequence converging to the limit $a \ne 0$ and let $\{n_k\} \subset \mathbb{N} $ be a divergent sequence of natural numbers. I want to show that the sequence $\{x_k n_k\}$ is ...
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If we have a sequence of measurable functions that is Cauchy with respect to the weak L^p norm, is it Cauchy with respect to the L^p norm?

If $(f_n)$ measurable on $(X,\mathcal{M},\mu)$, $f_n$ Cauchy with respect to weak quasi $L^p$-norm: $[f_n]_p=\sup_{\alpha>0}\alpha \lambda_{f_n}(\alpha)^{\frac{1}{p}} $ where $\lambda_{f}(\alpha)...
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Is the sequence converging?

Consider the sequence x1, x2, x3 ... defined by Xn={3+(1/n)} (if n is odd) and {4+(1/n)} (if n is even) As n→∞, the value of the first sequence will be 3 and the second one will be 4. Does this mean ...
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Cauchy sequence and boundedness

We know that every Cauchy sequence is bounded. and the converse may not be true. but if we define a sequence $x_n=n$ with metric $d(m,n)$=$\lvert \dfrac{1}{m}-\dfrac{1}{n}\rvert$. then this is a ...
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Limit evaluation using algebra of sequences and sequence theorems

By making use of only the theorems on sequences (ex: algebra of sequences/cauchy's first theorem of sequences/limit of geometric mean of a sequence etc), how to prove the following: $lim_{n\to\infty}(...
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Proof that the metric space $M$ is complete if every closed ball of $M$ is complete.

Let $M$ be a metric space I'm asked to prove the statement "Every closed ball of $M$ is complete $\implies$ $M$ is complete". My attempt at this is as follows: Let $\{y_i\}$ be a cauchy ...
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Show that sequence is a Cauchy sequence

Prove that given sequence $$\langle f_n\rangle =1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{(-1)^{n-1}}{n}$$ is a Cauchy sequence My attempt : $|f_{n}-f_{m}|=\Biggl|\dfrac{(-1)^{m}}{m+1}+\...
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Proving that $Y$ is complete

Let $X$ be a dense subset of $Y$. Let every cauchy sequence in $X$ converge to a point in $Y$ $(1)$. By the definition of dense I know that every point in $Y$ is either in $X$ or is a limit point ...
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Prove that a Cauchy sequence in $\mathbb{R}^n$ is convergent. [duplicate]

I need some help with this problem, I've seen that there is a similar problem about proving that Euclidean Space is a complete metric space, but I haven't learn about Euclidean or metric spaces and ...
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The space of all finite-degree polynomials $\mathbb{P}$ is not complete in any norm. [closed]

Let $\mathbb{P}$ denote the space of all finite degree polynomials in one variable. Show that $\mathbb{P}$ is never complete with respect to $P_1$ norm, i.e., $\|\cdot\|_1$ by giving an example ...
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Show that a sequence is Cauchy in $d_1$ iff it's Cauchy in $d_2$

Suppose two distances on $M$ are strongly equivalent, meaning that there exists $k$ and $K$ such that $$d_1(x,y) \le k*d_2(x,y); d_2(x,y) \le K*d_1(x,y)$$.Show that a sequence in $(M, d_1)$ is Cauchy ...
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Iterating a sequence and verifying its convergence

I am given a sequence $(f_n)_n$ where $n\in N$. $f_n : \Re \rightarrow \Re: x \mapsto 1$ $f_1:\Re \rightarrow \Re$ is defined as follows $$f_1 (x) = 1 + \int_0^x f_0 (t) dt$$ One sees that the ...
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Cauchy sequence by iterated Lipschitz functions

Let $(E,\delta)$ be complete, separable metric space and consider a sequence $\{\phi_n:n\in\mathbb{Z}\}$ of stationary ergodic Lipschitz maps defined of a probability space $(\,\Omega,\mathcal{A},\mu\,...
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Prove that $\Bbb Q$ is dense in $\Bbb R$ constructed by Cauchy sequences

Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! Let $\mathcal{C}$ be the set of Cauchy sequences of rationals. We define an ...
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Prove that three following statements regarding the complete linear ordering of Cauchy sequences are equivalent

Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! Let $\mathcal{C}$ be the set of Cauchy sequences of rationals. We define an ...
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Prove that the system of real numbers is complete by Cauchy sequence

Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help! Let $\mathcal{C}$ be the set of Cauchy sequences of rationals. We define an ...
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3answers
40 views

Show that sequence is Cauchy

I need to show that the above sequence is a Cauchy sequence. However, when using the definition of a Cauchy sequence, I get that $s(n) - s(m)$ is equal to some complicated sigmal notation expression, ...
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Linear operator is compact if and only if its adjoint is compact

Let $H$ be a Hilbert space, and $A:H\rightarrow H$ a linear operator. Prove that $A$ is compact if and only if $A^*$ is compact. I saw the following proof in my book - What I don't understand ...
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Confusion about Cauchy sequences

We know that a sequence of real numbers $(x_n)_{n\ge 1}$ is a Cauchy sequence if $\forall \epsilon>0$ $\exists n(\epsilon)$ so that $|x_{n+p}-x_n|<\epsilon$ $\forall n \ge n(\epsilon)$ and $p \...
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How to prove convergence using cauchy sequences

Let $(a_n)$ be a monotone and bounded sequence such that $a_n \to a$. Let $(b_n)$ be defined as $b_n = (a_1 + a_2 + ... + a_n)/n$. I know $(b_n)$ is monotone and bounded, but how do I prove that $b_n \...
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Solving a sum using Cauchy's sequence

could I get some help proving that the sum of this expression is equal to $m$ using Cauchy's criterion? $ 1+\frac{m-1}{m}+(\frac{m-1}{m})^{2}+(\frac{m-1}{m})^{3}+(\frac{m-1}{m})^{4}+....+(\frac{m-1}{...
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$c_{00}$ is not complete

I try to show that the space $c_{00}=\{(x_n):x_n=0 \text{ all but finitely many }n\}$ is not complete with respect to the norm $\|x\|_\infty=\max |x_n|$. My attempt: Let $(z_n)=\left(1,\frac{1}{2},\...
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1answer
52 views

How to prove every Cauchy Sequence in $\mathbb{R}^n$ converges

This is an analysis exercise that I have been struggling with for some time now. I am not familiar with metric spaces. In $\mathbb{R}$, the book that I am using proves this fact by showing that ...
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2answers
30 views

Understanding why an inequality holds in a proof

I was working on a real analysis problem and the following problem was one that I struggled with. Here are some preliminary definitions: A sequence of of points $\{u_{k}\}$ in $\mathbb{R}^{n}$ is ...
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Does Cauchy Completeness imply the Heine-Borel theorem generally?

I've been working through some reverse math with the completeness definitions of a metric space. More over, I've learned that in a metric space X that is ordered, The Least Upper Bound Property, ...
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1answer
55 views

If $\sum\limits^{\infty}_{n=1}|x_n|^2<\infty $ then $\{x_n\}$ is a Cauchy sequence

Let $\{x_n\}$ be a sequence of real numbers. How do I prove that if $\sum_{n=1}^\infty|x_n|^2<\infty$ then $\{x_n\}$ is a Cauchy sequence ? Is the converse true? How do I prove this? For the ...
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1answer
25 views

Integral Approximation to Infinite sum Vs Cauchy's first theorem

$$ \lim_{n\to \infty}\sum_{i=0}^n 1/(3*n +i) $$ . After applying cauchy's first theorem on pints, I get the answer as 1/4 , but after expressing the above sum as a definite integral I get the answer ...
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Prove that $\preccurlyeq$ is a linear ordering over the set of all equivalence classes of Cauchy sequences of rationals

Let $\mathcal{C}$ be the set of Cauchy sequences of rationals. We define an equivalence relation $\sim$ on $\mathcal{C}$ by $$(a_n) \sim (b_n) \iff \forall \epsilon >0, \exists N, \forall n>N: |...