Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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372 views

Cauchy sequence $a_n = 1 + \frac12 + \frac14 + … + \frac{1}{2^n}$

For the sequence $a_n = 1 + \frac12 + \frac14 + ... + \frac{1}{2^n}$, $n \ge 1$, find a formula $N = N(\epsilon)$ such that for all $\epsilon > 0$ and for all $m,n \ge N(\epsilon)$, $|a_m - a_n| &...
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2answers
179 views

Laurent Series, Cauchy, Pole Order

http://gyazo.com/8ef04b854bc3bbfb6b55a9af45e51fdc.png Since $f(z)$ not differentiable at $0$, isolated singularity at $z = 0$. By expanding the Laurent series and looking at the first term, I got a ...
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1answer
4k views

Convergence of Cauchy sequence $(x_n)$, where $x_n=1/n^2$.

I'm just looking at an example where we're asked to prove that the sequence $(x_n)$ is Cauchy, where $x_n=1/n^2$. (the example is from here at time 7:15 minutes; this requires registration but is free)...
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1answer
1k views

Bounded Inverse Theorem

$A$ is a bounded linear operator from $X$ to $Y$ (both Banach spaces). Show that if there exists $k > 0$ such that $\|Ax\| \geq k\|x\|$, for all $x$ then $\operatorname{range}(A)\,$ is closed. My ...
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1answer
245 views

Confusion regarding the proof that every Cauchy sequence is bounded

Here is the classic proof that Cauchy sequences are bounded: Let $\{x_n\}$ be a Cauchy sequence. For $\epsilon > 0$, choose $N$ such that if $n, m \geq N$, then $|a_n - a_m| < \epsilon$. Then $$...
15
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1answer
6k views

If a subsequence of a Cauchy sequence converges, then the whole sequence converges.

Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon > 0$. Take $N >...
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1answer
414 views

Issue with proof: Cauchy Completeness of Real Numbers

Having trouble understanding a cardinality-related argument when proving that all Cauchy sequences of reals numbers converge to a real limit. Came across it on CC Pugh's Real Mathematical Analysis, ...
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2answers
194 views

Show that sequences satisfying these two conditions need not be and must be Cauchy, respectively

Given a sequence $(a_n)$ prove that a) If it is known that $|a_{n+1}-a_n|< 1/n$ for all $n$, show that $(a_n)$ need not be a Cauchy Sequence. b) If it is known that $|a_{n+1}-a_n| < 1/2^n$ ...
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1answer
313 views

Cauchy sequences involving geometric series

let $\{x_n\}$ be a sequence of real numbers s.t $|x_{n+1} - x_n| \leq \alpha ^n$ $\forall n \in \mathbb{N}$ where $0 < \alpha < 1$ a) prove $|x_m - x_n | \leq \dfrac{\alpha^n}{1-\alpha} $ for $...
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2answers
185 views

Bound on the distance between two consecutive terms of a sequence guaranteeing the Cauchy condition

Let $0 < a < 1$ and let $(a_n)$ be a sequence in $\mathbb{R}$ such that $|a_{n+1} - a_n| < a^n.$ I want to show that $(a_n)$ is Cauchy. My idea was to bound $$ |a_{n+k} - a_n| \leq \sum_{i=...
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1answer
372 views

Show that sequence converges pointwise to a function that is not Riemann Integrable.

This is an exercise of the course of Measure and Integration and I'm having trouble to solve this. I not know how to show the sequence is of Cauchy and why are not R-Integrable. Let the sequence of ...
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1answer
263 views

Describe all the convergent and Cauchy sequences in this metric space

Consider the set of natural numbers $\mathbb N$ with the metric $$d(m,n)=\frac{\left|m-n\right|}{1+\left|m-n\right|}$$ Describe all convergent sequences and all Cauchy sequences in this metric space....
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407 views

Proving existence of limit

What is a way to prove the existence of a limit of the difference of two Cauchy sequences? What is a general definition that can be used to prove that a limit exists?
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1answer
67 views

Sum of countable Linearly independent vectors

Is $u=\sum_{n=1}^{\infty}\frac{1}{2^{n}}e_n$ an element of X, where $\{e_{n}\}_{n=1}^{\infty}$ is a maximal set of linearly independent vectors in X and X is a Banach space? In other words, are the ...
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3answers
199 views

Double Limit implies Successive Limits

I know it seems very stupid question, but is it right that: Suppose $X$ being a complete metric space. Then: $$\lim_{(m,n)}x_{(m,n)}\quad\text{exists} \quad\Rightarrow\quad \lim_n\lim_m x_{(m,n)}\...
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1answer
160 views

Cauchyness vs. Double Limits

Maybe there are some textbooks which might treat cauchyness by taking double limits... My question: Is it sufficient and necessary to consider the double limit: $$x_n\quad \text{cauchy}\quad \...
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1answer
527 views

Calculate cauchy product of series

These are the series I need to find the Cauchy product to: $$\sum_{n=0}^\infty q^n$$ and $$\sum_{n=0}^\infty nq^n$$ Is it just $$\sum_{j=0}^\infty\sum_{k=0}^j q_k^nnq^{n_ {j-k}}$$ or what am I ...
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4answers
197 views

Proving $\displaystyle\lim_{n\to\infty}a_n=\alpha$ $a_1=\frac\pi4, a_n = \cos(a_{n-1})$

Let $a_1=\frac\pi4, a_n = \cos(a_{n-1})$ Prove $\displaystyle\lim_{n\to\infty}a_n=\alpha$. Where $\alpha$ is the solution for $\cos x=x$. Hint: check that $(a_n)$ is a cauchy sequence ...
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1answer
548 views

Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to $...
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3answers
1k views

Difference of the Cauchy Sequences is Cauchy

Assume $\{a_n\}\;$ and $\;\{b_n\}\;$ are Cauchy sequences. Use a triangle Inequality Argument to prove $\{c_n=|a_n-b_n|\}\;$ is Cauchy. So in the answer key, the author proved it by taking $|c_n-c_m|...
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1answer
6k views

Prove that the Set of Bounded Linear Operators is Banach

Let $B(V,V')$ be the vector space formed by set of linear operators $T:V\rightarrow V'$. where $V,V'$ are normed vector spaces. Equip $B(V,V')$ with the norm $$ \|T\|=\sup\frac{\|T(x)\|}{\|x\|} $$ ...
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1answer
99 views

Question about Cauchy sequence convergence, and the idea behind it

Let there be $\epsilon>0$. $\{a_n\}$ is a Cauchy sequence if there is an index $k$, so that for every $n\ge k$ and for every $p \in \mathbb N$, $$|a_{n+p}-a_n|\lt \epsilon$$ And here is an ...
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1answer
290 views

How to find closed form formula for a sum

I am a PhD student in electrical engineering. I need to find a closed form formula for the following series: $$\sum_{k=1}^{\infty}\frac{1}{2}A_k^2e^{-k^2\sigma_m^2}(e^{k^2\sigma_m^2}-1)$$where $A_k= \...
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1answer
170 views

Convergence of Sequence of Real Numbers

Define a sequence of real numbers recursively as follows. Let $a_1 = 1$ and $a_{n+1} = 1 + \frac{1}{1+a_n}$. First, show the sequence is not monotonic. Second, show that $a_n \geq 1$ for all $n$ and ...
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2answers
73 views

Prove that $ \left(a_{n}\right)_{n=1}^{\infty} $ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $ 0<q<1 $

I'm stuck on a homework question, and could really use some help. Here is said question: "Assume that for every $n$ the following occurs: $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ when $ 0<q<1 $ ...
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1answer
145 views

Prove that the sequence $1+\sum_{k=1}^{n}\frac{k+1}{3^{k}+1}$ converges using Cauchy

I need some help with a homework question i'm having difficulty with. Here is the question: "Use the definition of cauchy sequence to prove that the series $\left(1+\frac{2}{3+1}+\frac{3}{9+1}+\cdots+...
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1answer
564 views

Proving completeness in normed vector space

Given norm $$ \| \{ a_n \} \| =\sum_{i=1}^\infty | a_n|,$$ where $E$ is a vector space of absolutely convergent series of real numbers under pointwise operations. How would one prove completeness of ...
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1answer
126 views

Sum of distances of a sequence is bounded $\longrightarrow$ sequence is Cauchy

A sequence $\{c_n\}$ in a metric space $(X,d)$ $(euclidean$ $distance)$ satisfies the following condition: There exists a positive real number $R\in \mathbb {R_{+}}$ such that for all $n\in \mathbb {...
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1answer
151 views

Completeness of metric spaces

edit: Another question about formatting. My questions on math.stackexchange keep cutting off the last few lines of my post. How do I fix this? The remaining lines show up in my edit box but not in the ...
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2answers
761 views

Decimal expansion of a Cauchy sequence

In one of the construction of $\mathbb{R}$ we make each real number an equivalence class of Cauchy sequences in $\mathbb{Q}$. More precisely, two Cauchy sequences $a_n$ and $b_n$ are equivalent iff $|...
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2answers
3k views

Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$

Let $E \subset F \subset X$ and $f:E\rightarrow Y$. We say that the function $g:F\rightarrow Y$ is an extension of $f$ if $g(x) = f(x)$ for all $x \in E$. Let $f: (a, b) \rightarrow \mathbb{R}$. ...
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1answer
182 views

Cauchy sequence with respect to Hausdorff metric

We know that if $(X,d)$ is a complete metric space, then $(CB(X),H)$ is complete too, where $CB(X)$ is the collection of non-empty closed bounded subset of $X$ and $H$ is the Hausdorff metric induced ...
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1answer
184 views

proving that $S_n$ is Cauchy.

$$S_n = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ... + \frac{(-1)^{n+1}}{2n-1} $$ Show that $(S_n)$ is a Cauchy sequence and hence that it converges to limit $L$. Show that $\frac{2}{3} < L &...
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2answers
187 views

Show that $x_n=\sum_{k=3}^n \frac{1}{k\ln^4(\ln k)}$ is Cauchy sequence.

I have the following sequence: $$ x_n = \sum_{k=3}^n \frac{1}{k\ln^4(\ln k)} $$ And I have to prove that this sequence is Cauchy sequence, i.e $\forall \varepsilon>0 \; \exists N: \forall n \ge N, \...
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1answer
571 views

Cauchy Sequence if $|s_{n+1} - s_n| < 2^{-n}$

Let $s_n$ be a sequence such that $|s_{n+1} - s_n| < 2^{-n}$ for all $n \in N$. Prove $s_n$ is a Cauchy sequence and hence a convergent sequence. Here's what I've started with: Proof: Take $\...
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2answers
73 views

Show that $X_{n}=\sum_{k=0}^{n} \frac{1}{k!}$ and $X_{n}=\sum_{k=0}^{n} \rho^{k}$ are Cauchy sequences.

Show that $X_{n}=\sum_{k=0}^{n} \frac{1}{k!}$ and $X_{n}=\sum_{k=0}^{n} \rho^{k}$ are Cauchy sequences. I know that for each sequence I have to show the following: "$\forall \epsilon>0 \text{ }...
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2answers
4k views

Show that every monotonic increasing and bounded sequence is Cauchy.

The title is kind of misleading because the task actually to show Every monotonic increasing and bounded sequence $(x_n)_{n\in\mathbb{N}}$ is Cauchy without knowing that: Every bounded non-empty ...
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2answers
343 views

Show that this sequence converges. (cauchy criterion)

Given $a_0 \geq 0$ and a sequence ($a_n)_{n\in\mathbb{N}}$ $$ a_{n+1}= \frac1{(2+a_{n})}.$$ for ${n\in\mathbb{N_0}}$. Show that $(a_n)_{n\in\mathbb{N}}$ is convergent and determine the limit. ...
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2answers
4k views

Cauchy sequences and uniformly continuous functions

I'm trying to prove that if $\{x_n\}$ is Cauchy sequence which located in $E$ ($f$ is uniformly continuous) then $\{f(x_n)\}$ is a Cauchy. let say {$x_n$} is a Cauchy sequence in $E$ and $E \to R$ ...
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2answers
778 views

Equivalence of cauchy sequences/completeness in topological spaces.

In general a topological space may not have an equivalent of a metric and its not possible to define completeness of a sequence that way. An alternative that I was thinking about was to say a sequence ...
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2answers
5k views

Metric space is totally bounded iff every sequence has Cauchy subsequence

Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence. I think I proved the Cauchy subsequence part: Let $a_{0},a_{1}, a_{2}, a_{3}, a_{4},...\in X$ be ...
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2answers
604 views

Why is the sequence $ a_n = \left(1+\frac{1}{n}\right)^n $ Cauchy?

I was looking at the post: Cauchy Sequence that Does Not Converge And the top answer was this sequence: $ a_n = \left(1+\frac{1}{n}\right)^n$. I understand that this sequence converges to $e$, ...
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3answers
126 views

Cauchy convergent sequences

Suppose that $(a_n)$ and $(b_n)$ are convergent sequences and that $b_n > 0$ for all $n$. Is it true that $(a_n / b_n)$ is Cauchy? If it is true, prove it. If it is not true, give a counterexample ...
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1answer
472 views

$f$ uniformly continuous , $a_n$ Cauchy $\Rightarrow f(a_n)$ is Cauchy [closed]

Let $I$ be an interval and let $f: I\to \mathbb{ R}$ be uniformly continuous on I. Suppose that $\{a_n\}$ is a Cauchy sequence in $I$. Prove that $\{f(a_n)\} $is a Cauchy sequence.
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131 views

Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$

How do I prove: Let $\lim\limits_{n\to \infty} a_n = \infty$ and $\lim\limits_{n\to \infty} b_n = \infty$ Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$ Thank you
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1answer
73 views

Cauchy sequences in two cases in two space metrices

Is it $f_n(x) = x^n $ cauchy sequence if: $d(f,g) = \int_0^1|f(x)-g(x)|dx$ $d(f,g) = \mbox{sup}_{x \in [0,1]}|f(x)-g(x)|$ What, if $f_n(x) = x^n - x^{2n}$ ? In first case we have $f_n(x) = x^n $ ...
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3answers
117 views

Existence of non-convergent sequence with first differences converging to zero [duplicate]

Does there exist a sequence $x_n$ such that $|x_n - x_{n+1}| \rightarrow 0$ but $x_n$ isn't convergent? I was looking for such a sequence, but I can't find one. Maybe if $|x_n - x_{n+1}| \rightarrow ...
2
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0answers
2k views

Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{ 2^2}+\cdots+\frac{1}{n^2}$

Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{2^2}+\frac{1}{3^2}+ \ldots +\frac{1}{n^2}$ My attempt Take $|x_m-x_n|$, where $m>n$, We have $|x_m-x_n|=\frac{1}{(n+1)^...
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2answers
505 views

Prove that open subspace of a topologically complete space is topologically complete

I'm trying to prove that an open subspace of a topologically complete space is topologically complete. I follow the hint in the book. We defined $\phi : U \rightarrow R$ by the equation $$\phi(x) =...
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1answer
248 views

Revisiting a problem on Cauchy sequences

I posted earlier this week Proving a sequence is Cauchy given some qualities about the sequence I believe I solved the question myself, but my roommate has a different method of solving it, and is as ...