Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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Let $(a_n)$ be a sequence such that $\lim_{N\rightarrow \infty} \sum_{n=1}^N |a_n - a_{n+1}| < \infty$. Show that $(a_n)$ is Cauchy.

Let $(a_n)$ be a sequence such that $\lim_{N\rightarrow \infty} \sum_{n=1}^N |a_n - a_{n+1}| < \infty$. Show that $(a_n)$ is Cauchy. Proof: We know $(x_n)$ is Cauchy if $\forall\ \epsilon > 0, ...
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56 views

Let $(x_n)$ be Cauchy with a subsequence $(x_{n_k})$ such that $\lim_{k\to\infty} (x_{n_k}) = a$. Show that $\lim_{n\to\infty} (x_{n}) = a$

Let $(x_n)$ be Cauchy with a subsequence $(x_{n_k})$ such that $\lim_{k\rightarrow\infty} (x_{n_k}) = a$. Show that $\lim_{k\rightarrow\infty} (x_{n}) = a$. Proof: Since $(x_n)$ is Cauchy, we know it ...
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45 views

Convergence of recursive sequence involving cosine

Let x ∈ R. Consider the sequence ${a_n}$, where $a_1$ = x and $ a_{n+1 }= cos(a_n)$. From picture I can observe that ${a_n}$ converges. But how to prove it analytically. I tried to show it is ...
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52 views

Question about Cauchy sequences are convergent in $\mathbb{R}^k$

Let $\{p_n\}$ be a Cauchy sequence in $\mathbb{R}^k$. Let $K = cl \{p_n\}$ be an infinite compact subset of $\mathbb{R}^k$ (so $K$ is the closure of the set that consists of all distinct elements in ...
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21 views

Checking out my demonstration on metric spaces

Let $(X,d)$ a metric space. Prove that $(X,d)$ is complete if, and only if, for all sequence of closed embedded non-empty sets $(F_{n})$ in $X$ such that $\lim_{n\rightarrow \infty}\textrm{diam}(F_{n})...
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84 views

Let $f_n \in L_p(X, \mathbb{X}, \mu)$, $1 \leq p < \infty$, and let $\beta _n$ defined for $E \in \mathbb{X}$ by

Let $f_n \in L_p(X, \mathbb{X}, \mu)$, $1 \leq p < \infty$, and let $\beta _n$ defined for $E \in \mathbb{X}$ by $\beta _n = (\int_E |f_n|^pd \mu)^{\frac{1}{p}}$ and suppose that $f_n$ is a ...
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1answer
68 views

If $\sum x_n$ converges, is the even partial sums of the sequence squared Cauchy?

If $\sum x_n$ converges, is the sequence $a_k =\sum_{n=1}^k x_{2n}^2$ Cauchy? I suspect that it isn't but a bit stuck on finding a counter example. For a valid counterexample, I believe I need a ...
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51 views

Understanding a Cauchy Sequence Proof

I've got some general confusion over Cauchy sequences I'm trying to clear up. Intuitively, I know that the Cauchy theorem lets us prove convergence of a sequence without knowing about the real number ...
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144 views

Show that a recursive sequence $(x_n)$ is Cauchy

The given sequence is defined as $x_1 = 0$, $x_2 = 1$ and $x_{n+2} = \frac{1}{3}x_{n+1} + \frac{2}{3}x_n$ for $n \geq 1$. I seek to show that it is Cauchy. So how I planned on showing this was to ...
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20 views

Determine if subset $S$ of $\mathbb{R}$ is complete

Given $S = \{x \in \mathbb{R} \setminus\mathbb{Q}\mid x^2 \leq \frac{1}{4} \} \subset \mathbb{R}$, determine if it is complete. Given that a subset $S$ of $\mathbb{R}$ is $\textit{complete} $ if ...
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37 views

Prove the sequence is Cauchy

Prove with definition that it is Cauchy $$a_n = \frac{n+3}{2n+1},$$ wheree $n$ is a natural number I have seen other examples such as $\frac{1}{n}$ and such that show how to prove they are Cauchy ...
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39 views

Prove that the following set $A = \{x \in \mathbb{R} \setminus\mathbb{Q}\mid x^2 \leq \frac{1}{4} \} \subset \mathbb{R}$ is complete.

A subset $S$ of $\mathbb{R}$ is complete if every Cauchy sequence consisting of elements of $S$ converges to an element of $S$. Prove that the following set $A = \{x \in \mathbb{R} \setminus\mathbb{Q}...
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31 views

proving convergence by showing a sequence is Cauchy

Let $(x_n)_{ n=1}^{∞} $ be a sequence of real numbers such that $|x_{n+1} − x_n| ≤ \frac {1} {2^n}$ for all $n$. Show that $(x_n) ^∞ _{n=1}$ converges. It seems pretty clear that I can prove that ...
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32 views

C($\mathbb{R}$) is complete under the discrete metric.

Examine if the space $X=C(\mathbb{R})$ endowed with the discrete metric $d_0$ is complete or not. We know that $(X,d_0)$ is complete if every Cauchy sequence is finally constant. We want to construct ...
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28 views

proving limits of sequences using cauchy criterion

The question is as follows: Let $\{s_n\}^∞_1$ be a sequence of real numbers such that $s_n > 0$ for $n > k$. Suppose $s = lim_{n→∞}s_n$ exists and is a finite real number. Prove that $s ≥ 0$. ...
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47 views

Triangle Inequality for proving Cauchy Criterion for Series

I need to show that the statement $$\forall \epsilon > 0, \text{there is an } N\in \mathbb{N} \text{ such that for all } n\geq N, |\Sigma _{k=n+1}^{\infty} a_k | < \epsilon$$ implies $$\forall ...
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55 views

Cauchy sequences always have a largest or smallest element past an arbitrary index

The problem is as follows: Let $(x_n)_n$ be Cauchy. Show that either $$ (1)\; \forall N\in\mathbb{N}, \exists\bar{n}\geq N \text{ s.t. } \forall n\geq N, x_n\leq x_{\bar{n}} $$ or $$ (2)\; \forall ...
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13 views

Clarification on how a particular space is defined

I am tasked with proving that the the metric space $(C[0,1], d)$ is complete where $d$ is the infinity-metric. As of right now, I interpret the set $C[0,1]$ as the set of continuous functions from $...
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44 views

For a sequence, how should we check that the distance between two consecutive terms $a_{n+2}$ and $a_{n+1}$ is less than $a_{n+1}$ and $a_n$?

Background It can be proved that for $ 0 < \alpha <1$, if $\{a_n\}$ is a sequence which satisfies $$ |a_{n+2} - a_{n+1}| \leq \alpha |a_{n+1} - a_n|$$ Then $a_n$ is a Cauchy ...
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What sequences are Cauchy in all metrics for a given topology?

Different metrics for the same topology can have different sets of Cauchy sequences. But I'm interested in what sequences are Cauchy in every metric for a given topology. For a completely metrizable ...
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136 views

Why can't completeness be defined on topological spaces without using metrics?

I have heard it said that completeness is a not a property of topological spaces, only a property of metric spaces (or topological groups), because Cauchy sequences require a metric to define them, ...
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35 views

Let $s_{0}$ and $s_{1}$ be arbitrary and $s_{n+1}$=($s_{n} $+ $s_{n-1}$ ) /2 n>=1 show Cauchy [duplicate]

Let $s_{0}$ and $s_{1}$ be arbitrary and $s_{n+1}$=($s_{n} $+ $s_{n-1}$ ) /2 n>=1 show Cauchy show {$s_{n}$} is a Cauchy sequence. Conclude that {$s_{n}$} is convergent.
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25 views

Determine using only the definition if it is a Cauchy sequence

Determine using only the definition if it is a Cauchy sequence $(2n^2+1)/n^2$ Where did I go wrong? This is what I have so far: $$\frac{2m^2+1}{m^2}-\frac{2n^2+1}{n^2}= \frac{n^2(2m^2+1)}{m^2n^2}-\...
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86 views

True of False: Every sequence contains a Cauchy subsequence?

For my practice midterm exam this is a question: State whether the following are true or false AND briefly explain why (or give counterexample): Statement: Every sequence contains a Cauchy ...
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1answer
53 views

Show that on any bounded subset $K\subseteq \Bbb{C}$, the sequence is a Cauchy with respect to the norm $\|\cdot\|_{\infty,K}$

Define $S_N (z)=\int_{1}^{N}t^z e^{-t}dt$ and $f(z)=\int_{1}^{\infty} t^z e^{-t}dt$. Show that on any bounded subset $K\subseteq \Bbb{C}$, we have $S_N$ is a Cauchy sequence with respect to the norm ...
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49 views

Prove that $[0,\pi/2]$ is $d$-complete, where $d(x,y)=\left|\sin(x-y)\right|$.

Let $d$ be the metric on $[0,\pi/2]$ defined as: $$d(x,y)=\left|\sin(x-y)\right|.$$ Prove that $[0,\pi/2]$ is $d$-complete. Attempt. Let $(x_n)$ be a $d$-Cauchy sequence on $[0,\pi/2]$, meaning ...
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1answer
67 views

How to prove that it is a Cauchy sequence?

Let $(a_n)\in\mathbb{R}^{\mathbb{N}}$ such that : $\forall (p,q)\in \mathbb{N}^2$, $\vert a_{p+q}-a_p -a_q \vert \le 1$. Show that $(\frac{a_n}{n})_{n\in \mathbb{N}^*}$ is a Cauchy sequence (without ...
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1answer
31 views

Diameter hypothesis of Cantor's Intersection Theorem on a complete metric space.

Cantor's Theorem states: "Suppose that X is a non-empty complete metric space, and $C_n$ is a sequence of closed nested subsets of X whose diameters tend to zero: $\lim_{n \rightarrow \infty} diam(...
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105 views

Prove that if $ (x_n) $ has a Cauchy subsequence, then for any decreasing sequence

Let $(X,d) $ be a metric space and let $(x_n)$ be a sequence in $X$ . Prove that if $(x_n)$ has a Cauchy subsequence, then for any decreasing sequence of $\epsilon_k$ -> $0$, there is a subsequence $(...
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123 views

Question about Cauchy Sequence proof for $\sum_{i=1}^\infty \frac{1}{i^2}$

I was looking through a proof that the sum of $$\sum_{i=1}^\infty \frac{1}{i^2} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}...$$ is convergent. I know there are probably other ways to prove this, ...
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1answer
44 views

Sequence defined as sum.

Let $a_n=\sum_{k=0}^{n-1}\frac{(-1)^{k}}{k!}$. Prove that this sequence is Cauchy. I know that this is the partial sum of the series $\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}$ wich converges, but I don't ...
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224 views

Using Cauchy convergence to prove that $\sum_{k=1}^{n} z_{k}$ converges if $\sum_{k=1}^{n}\left | z_{k} \right |$ does

Consider the sequence $\left \{ z_{k} \right \}$ in $\mathbb{R}.$  Let the sequence  $$\left \{ x_{n} \right \} = \sum_{k=1}^{n}z_{k}\ \ \ \   \text{and}\ \ \  \left \{ y_{n} \right \} = \sum_{k=1}^{n}...
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111 views

Cauchy sequences - Infinite sums

Let $\{x_n\}$ be a sequence of real numbers. Suppose that for each $\epsilon > 0$ there is an $N \in \mathbb{N}$ such that $m \geq n \geq N$ implies $ |\sum_{k=n} ^m x_k | < \epsilon$. Prove ...
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1answer
30 views

If for all $n \in N$, $|a_n| < 2$ and $|a_{n+2} - a_{n+1}| \leq \frac{1}{8}|a_{n+1}^2 - a_{n}^2|$ .

Suppose that ${a_{n}}$ is a sequence such that, for all $n \in N$, $|a_n| < 2$, and $|a_{n+2} - a_{n+1}| \leq \frac{1}{8}|a_{n+1}^2 - a_{n}^2|$ . Prove that ${a_n}$ is a Cauchy sequence. My ...
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124 views

Is the sequence $a_{n} = \frac {1}{2^2} + \frac{2}{3^2} + … +\frac{n}{(n+1)^2}$ Cauchy?

First: I tried substituting natural numbers for $n$ to calculate the consecutive terms of the sequence and then see the difference between their values and I found that the difference is decreasing ...
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49 views

Counterexample: Cauchy condition without absolute value for convergence of a sequence

Let $(a_n)$ be a sequence of non-negative real numbers, not necessarily increasing. The Cauchy criterion states that $(a_n)$ converges if and only if $\forall \epsilon >0$, there exists an $N$ ...
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91 views

Cauchy sequence where the set of outputs is finite

If $\{a_n\}$ is a Cauchy sequence, and $S = \{a_n |n\in\mathbb{N}\}$ is finite, then $\{a_n\}$ is constant from some point on. The statement makes sense, but I'm not quite sure how to start. I feel ...
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83 views

Is the sequence $a_{n} = 1 + \frac14 + \frac{2^2}{4^2} + \cdots +\frac{n^2}{4^n}$ Cauchy?

I think that it is Cauchy (but I am not sure of this) and this is my proof: $$|a_{m} - a_{n}| = \left|\frac{n+1}{4^{n+1}} + \frac{n+2}{4^{n+2}} + ..... + \frac{m^2}{m}\right| =\sum_{k=n+1}^{m} \frac{...
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42 views

Show that $\{e^{(k)}: k\ge 1\}$ is closed as a subset of $l_1$.

Let $e^{(k)}=(0, \ldots, 0, 1, 0, \ldots)$ where the $k$th entry is $1$ and the rest are $0$s. Show that $\{e^{(k)}: k\ge 1\}$ is closed as a subset of $l_1$. My attempt: A theorem in my textbook ...
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43 views

Cauchy Sequence Proof using Power series

Let {${x_n}$} be a sequence such that there exists a $0<C<1$, such that: $|x_{n+1}-x_n|\leq$$C|x_n-x_{n-1}|$. Prove that {$x_n$} is a Cauchy Sequence. I'm given a hint that $1+C+C^2...+C^n=\...
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80 views

Cauchy sequence exercise question (Exercise 2.6.4 Abbott analysis)

I have a problem solving the question below. I'm stuck at the stage (ii) of my solution below. Problem: Let $(a_n)$ be a Cauchy sequence. Decide whether the following sequence is a Cauchy sequence, ...
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45 views

There are no nontrivial metric spaces such that every sequence is a Cauchy sequence

Is my intuition correct here? If our metric space consists of a single point $x$, any sequence in it would just be $x, x, x, \dots$ which is obviously Cauchy. If we have any metric space $(M, d)$ ...
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383 views

Proving that a sequence is not Cauchy

I'm trying to prove that the sequence $\left(\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},\cdots\right)$ is not a Cauchy ...
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1answer
39 views

Proving this sequence is Cauchy

How can I prove that the sequence $\{x_k\}$ where $x_1 = 4, x_k = x_{k-1} + 4(\frac{1}{4k-3} - \frac{1}{4k-5})$ is Cauchy? I understand the definition of Cauchy and I can see that the sequence is ...
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126 views

Stephen Abbott “Understanding Analysis” p.81 Exercise 2.8.3.

I am reading "Understanding Analysis" by Stephen Abbott. I solved p.81 Exercise 2.8.3. But I am not sure that my answer is correct. Is the following answer correct or not? Let $\{a_{ij} : i, j \in ...
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23 views

On Cauchy sequences over finitely generated modules over complete DVR

I have a question on Cauchy sequences. Let $R$ be a complete DVR with a valuation $v$, $\pi$ a uniformizer, $V$ a finitely generated $R$-module. Since $R$ is PID, $V$ is the direct sum of cyclic ...
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2answers
30 views

LUB (if it exists) of a complete set belongs to that set: Validity

By LUB I mean the least upper bound of the set. And the definition of complete set I am using is that every Cauchy sequence in that set must converge in that set. So by these two assumptions. I cannot ...
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1answer
35 views

$c_n = ∑a_k a_{n-k}$ is convergent ot not

Let $a_n =\frac {(-1)^n}{(1+n)^{0.5}}$ and let $c_n = ∑a_k a_{n-k}$ Then will $\sum^\infty _{n=0} c_n$ be converge? If $\sum^\infty _{n=0} a_n$ converged absolutely I could say that the ...
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38 views

The existence of two non-intersecting intervals

Let $\{a_i\}$, $\{b_i\}$ be two non-equivalent Cauchy sequences. Prove that there exist two non-intersecting intervals $I_1$, $I_2$ such that almost all $\{a_i\}$ belong to $I_1$ while almost all $\{...
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3answers
57 views

Let $0 < a \leq 1$ and $s_1 = a/2$ , $2s_{n+1} = s_n^2 + a$ , Then how to show that the sequence is convergent.

Let $0 < a \leq 1$ and $s_1 = a/2$ , $2s_{n+1} = s_n^2 + a$ , Then how to show that the sequence is convergent. My Try : I have tried to find out $s_{n+1} - s_n$ and try to understand the ...