Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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Determining Complete Metric Spaces

I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$ My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that $\...
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1answer
117 views

Correctness of Analysis argument with Cauchy sequences

Let $(x_n)$ and $(y_n)$ be Cauchy sequences in $(X, d)$. Show that $(x_n)$ and $(y_n)$ converge to the same limit iff $d(x_n, y_n) \rightarrow 0$ Proof $\rightarrow$ Suppose $(x_n) \to a$ and $(y_n) ...
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1answer
307 views

Cauchy Sequences and Analysis

Let $(x_n)$ and $(y_n)$ be Cauchy sequences in a metric space $(X, d)$. Show that the sequence $(d(x_n, y_n))$ is a cauchy sequence in $\mathbb{R}$. What is the significance of $\mathbb{R}$ in this ...
2
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1answer
473 views

Determine whether metric spaces with metrics of the form $d(x,y)=|f(x)-f(y)|$ are complete

How to decide if the metric spaces $((0,1)$, $d(x,y)=|x^2-y^2|)$ and $((-\frac{\pi}{2},\frac{\pi}{2})$, $d(x,y)=|\tan x-\tan y|)$ are complete or not? For the first metric, I let any Cauchy sequence $...
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2answers
2k views

Need to prove $f$ continuous at $x_0$ iff for every monotonic sequence $(x_n)$ converging to $x_0$ we have $\lim f(x_n)=f(x_0)$

This was a problem that the Professor went over in class, but I am having trouble understanding and finishing the proof. The full question is: $f:I \rightarrow \mathbb R$ is continuous at $x_0 \in I$ ...
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2answers
4k views

Is the set of integers Cauchy complete?

http://en.wikipedia.org/wiki/Complete_metric_space says that a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, ...
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4answers
21k views

Cauchy sequence is convergent iff it has a convergent subsequence

Prove that if $\left ( x_{n} \right )$ is a Cauchy sequence in a metric space X then $\left ( x_{n} \right )$ is convergent if and only if $\left ( x_{n} \right )$ has a convergent subsequence. Note: ...
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2answers
209 views

Question in real analysis

I need help with this problem. Show that a Cauchy sequence in $[0,1]$ must converge to a point of $[0,1]$. Thank you
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1answer
139 views

Convergent & Cauchy Sequence related prove

(1) Consider the two convergent sequences $\{a_n\}$and $\{b_n\}$ such that $$\{a_n\}\to a$$ and $$\{b_n\}\to b$$ for $n\to\infty$. Prove that $$\{a_n+b_n\}\to a + b$$ for $n\to\infty$. (2) Prove ...
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1answer
2k views

Banach spaces and their unit sphere

Let $X$ be a normed vector space. Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete if ...
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2answers
4k views

Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$. Is $f$ continuous? Let $f$ be ...
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3answers
2k views

How do I find the limit of the sequence $a_n=\frac{n\cos(n)}{n^2+1}$ and prove it is a Cauchy sequence?

I need to study the limit behavior of $a_n=\frac{n\cos(n)}{n^2+1}$, which can be written as $\frac{n}{n^2+1}\cos(n).$ I knew that it wasn't going to be monotone because $cos(n)$ oscillates between -1 ...
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2answers
1k views

Continuity, uniform continuity and preservation of Cauchy sequences in metric spaces.

Let $ X $ and $ Y $ be metric spaces, and let $ f: X \to Y $ be a mapping. Determine which of the following statements is/are true. a. If $ f $ is uniformly continuous, then the image of every Cauchy ...
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1answer
12k views

How do I prove a uniformly continuous function preserves Cauchy sequences?

Let $f$ be a uniformly continuous function on A of $\Bbb{R}$. How do I show that if $a_n$ is Cauchy, then $f(a_n)$ is Cauchy. This is what I have worked on, but it does not quite make sense since I ...
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6answers
3k views

Is $x^n$ Cauchy in $(C[0, 1], \|\cdot\|_{\infty})$?

Consider the sequence of functions \begin{equation} f_n(x) = x^n, \quad x \in [0, 1]. \end{equation} Is this sequence Cauchy in $(C[0, 1], \|\cdot\|_{\infty})$? The pointwise limit is not ...
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3answers
678 views

Cauchy Sequence. What is this question actually telling me?

Let $(a_n)$ be a sequence such that $\lim\limits_{N\to\infty} \sum_{n=1}^n |a_n-a_{n+1}|<\infty$. Show that $(a_n)$ is Cauchy. So basically I am told that the sum of the difference isn't infinite. ...
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1answer
4k views

Continuous Functions and Cauchy Sequences

We know that if a function $f: A \mapsto \mathbb{R}$, $A \subseteq \mathbb{R}$, is uniformly continuous on $A$ then, if $(x_n)$ is a Cauchy sequence in $A$, then $(f(x_n))$ is also a Cauchy sequence. ...
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3answers
32k views

Proving that a sequence such that $|a_{n+1} - a_n| \le 2^{-n}$ is Cauchy

Suppose the terms of the sequence of real numbers $\{a_n\}$ satisfy $|a_{n+1} - a_n| \le 2^{-n}$ for all $n$. Prove that $\{a_n\}$ is Cauchy. My Work So by the definition of a Cauchy sequence, for ...
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1answer
1k views

Which of the following metric spaces are complete?

[NBHM_2006_PhD Screening test_Topology] Which of the following metric spaces are complete? $X_1=(0,1), d(x,y)=|\tan x-\tan y|$ $X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$ $X_3=\mathbb{Q}, ...
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2answers
394 views

Cauchy sequence alternative definition.

Is it possible to define a Cauchy sequence as follows? Let $(X,d)$ be a metric space and $(x_{n})_{n\in \mathbb{N}}$ be a sequence in it. Then $(x_{n})_{n\in \mathbb{N}}$ is Cauchy iff $\lim_{(j,k)\...
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1answer
746 views

Is the sequence $\{1/n^2\}$ Cauchy?

Is this sequence Cauchy? $$\left\{\frac{1}{n^{2}}\right\}$$ My attempt: Suppose that converges as it goes to $0$ and is therefore Cauchy, but I lack formality in my reply.
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1answer
519 views

Cauchy sequence under uniformly continuous function

We know that under uniformly continuous function a Cauchy sequence goes to a Cauchy sequence. Let $f:A\rightarrow \mathbb{R}^m$ be uniformly continuous function where $A\subseteq\mathbb{R}^n$, I need ...
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5answers
2k views

Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy? I know that Cauchy condition means that for each $\varepsilon>0$ there exists $N$ such that $d(x_p,x_q)<\...
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3answers
1k views

Cauchy Sequence in $X$ on $[0,1]$ with norm $\int_{0}^{1} |x(t)|dt$

In Luenberger's Optimization book pg. 34 an example says "Let $X$ be the space of continuous functions on $[0,1]$ with norm defined as $\|x\| = \int_{0}^{1} |x(t)|dt$". In order to prove $X$ is ...
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4answers
471 views

Alternate proof that a sequence is Cauchy

Is it sufficient to show that for any $\epsilon > 0$ that there exists $N$ such that if $n$ is greater than or equal to $N$, then $d(s_n, s_{n+1})$ is less than $\epsilon$ to prove that a sequence $...
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1answer
589 views

Cauchy sequence and convergence

A sequence is said to be Cauchy sequence if for given any integer n, there exists a positive real number R, such that for any n1, n2 > n, mod{n1th term - n2th term} 1,0,1,0,1,0,1,0........ Now for ...
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3answers
2k views

A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy

Give an example of a sequence $(x_n)$ of real numbers, where $\displaystyle\lim_{n\to+\infty}|x_n-x_{n+1}|=0$, but $(x_n)$ is not a Cauchy sequence
64
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5answers
11k views

Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?