Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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How to show that a Cauchy sequence of sequences converges

Let $c_0$ be the space of real-valued sequences $\{x_n\}$ which converge to zero, equipped with the metric $d(\{x_n\}, \{y_n\}) = sup_n |x_n − y_n|$. I want to show that the metric space $(c_0, d)$ is ...
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240 views

Finding a special subsequence of any Cauchy sequence

Let $(X,d)$ be a metric space and let $(x_n)$ be a Cauchy sequence in $X$. Let $(\epsilon_n)$ be a sequence of real numbers and decrease to $0$. Show that there is a subsequence $(x_{n_k})$ of $(x_n)$ ...
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3answers
513 views

Is this space a Hilbert Space?

I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way: $ x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t. $ Is this space a Hilbert ...
5
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1answer
148 views

Using the Uniform Cauchy Criterion theorem

Suppose $(f_n)$ is defined, continuous on $[a,b]$, and differentiable on this open interval. Then $c \in [a,b]$ and $(f_n(c))$ converge $(f'_n)$ and converge uniformly on $(a,b)$ respectively. How ...
5
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1answer
8k views

Proving convergent sequences are Cauchy sequences

Prove that if $x_n \rightarrow a, n \rightarrow \infty$ then $\{x_n\}$ is a Cauchy sequence. I believe I have found the proof as follows, wondering if there are any simpler methods or added ...
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0answers
645 views

Prove the space of bounded sequences is Banach

http://www.math.ucla.edu/~tao/resource/general/121.1.00s/exam1sol.pdf Here is a proof, but I cannot fully understand why it does not give a proof that $x$ is a bounded sequence (i.e. $x$ is in the ...
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204 views

Baby Rudin Exercise 4.13 Alternate Proof Verification

I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
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6answers
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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 ...
4
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3answers
1k views

Find a limit of the recursive sequence

The task is to prove sequence convergence and find a limit. $x_0=0$ $x_1=1$ $x_{n+1}=\frac {x_n + n \cdot x_{n-1}} {n+1}$ I have computed some values of a sequence to build up some idea of the data: ...
4
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3answers
135 views

How to prove that if $\sum _{n=1}^{\infty }a_n\:$ converges then $\sum _{n=1}^{\infty }a_na_{2n}\:$ converges?

How to prove that if $\sum _{n=1}^{\infty }a_n\:$ converges then $\sum _{n=1}^{\infty }a_na_{2n}\:$ converges? Note: $a_n \in \mathbb R$ I tried to prove it using cauchy criterion The idea was to ...
4
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3answers
146 views

To show that a recursively defined sequence by $x_1=\frac12$ and $x_{n+1}=\frac{x_n^3 + 2}{7}$ is Cauchy - How?

Show that the sequence defined by $x_1$ = $\frac{1}{2}$ and $x_{n+1} = \frac{x_n^3 + 2}{7}$ for $n \in N$ satisfies the Cauchy criterion. I don't understand how to go about this problem. Is it ...
4
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2answers
220 views

Banach space and sequence

Let $(X,\left\Vert \cdot \right\Vert)$ be a normed space. Show that $X$ is Banach space (under the given norm) if and only if the sum $\Sigma_{n=1}^{\infty}x_n$ converges in $X$ for any sequence $(x_n)...
4
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1answer
340 views

Are all inner product spaces over the field of the real numbers Hilbert spaces?

All Cauchy sequences over $\mathbf {R}$ converge. Does this mean every inner product space over $\mathbf {R}$ is a complete metric space? If not, what is an example a non-Hilbert inner product space?
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6answers
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If $\lim_{n \rightarrow \infty} (a_{n+1}-\frac{a_n}{2})=0$ then show $a_n$ converges to $0$. [duplicate]

I have been stuck on this question for a while now. I have tried many attempts. Here are two that I thought looked promising but lead to a dead end: Attempt 1: Write out the terms of $b_n$: $$b_1=...
4
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3answers
574 views

How can you prove $\{x_n\}$ is a Cauchy sequence if $|x_{n+1} – x_n| \le r|x_n - x_{n-1}|$ for some $0<r<1$?

Let $\{x_n\}$ be a sequence and let $r$ be a number such that $0 < r < 1$. Suppose that $$|x_{n+1} – x_n| \le r|x_n - x_{n-1}|$$ for all $n>1$. Prove that $\{x_n\}$ is a Cauchy sequence. I ...
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2answers
5k views

Examples of Incomplete Spaces [closed]

A metric space is complete if every cauchy sequence is convergent. To make space incomplete either i can change the metric or the ambient space. For example if I change real numbers into rational ...
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3answers
77 views

Proof $\text{Si}(n) $ is convergent

I am trying to prove that the sequence formed by the Si function, $\text{Si}(n) = \int_0^n \frac{\sin(u)}{u} \mathrm{d}u$, is convergent as $n\rightarrow \infty$. The only twist is the lower bound of ...
4
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2answers
346 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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3answers
682 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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3answers
70 views

Prove the geometric sequence $(r^n)$ is Cauchy if $|r|<1$

Suppose, towards a contradiction, that $(r^n)$ is not Cauchy. Then $\exists \epsilon >0$ such that for every $n\in \mathbb{N}$, $\exists m > n$ such that $|r^m - r^n| \geq \epsilon$. Then $|r^n|...
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1answer
74 views

specifying which sequences converge equivalent to specifying topology

I know in many cases, specifying which sequences converge is sufficient for specifying which topology is being used. I was wondering in which kinds of scenarios is this necessarily the case, and when ...
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3answers
262 views

Rate of convergence of summable sequence

Suppose $a_n$ is a nonnegative real sequence such that \begin{equation} \sum_n a_n <\infty. \end{equation} What do we know about $a_n$? We know $a_n\to 0$. We know $$\lim\inf n a_n = 0.$$ But can ...
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3answers
64 views

Showing sequence is Cauchy by Definition

Question: I have to show that sequence $(x_n)$ defined by $x_n=\frac{n+(-1)^n}{2n-1}$ , $n=1,2,3,...$. Is Cauchy sequence using definition only. My attempt: (I can see given sequence is Cauchy ...
4
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1answer
252 views

$|x_{n+1} - x_n| \le C |x_n - x_{n-1}|.$ Prove $(x_n)$ is cauchy

Let $\{x_n\}$ be a sequence such that there exists a $0 < C < 1$ such that $$|x_{n+1} - x_n| \le C |x_n - x_{n-1}|.$$ Prove that $\{x_n\}$ is Cauchy. Hint: You can freely use the formula (for $...
4
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1answer
62 views

Why does a “less than” expression become a “less than or equal to” expression after taking a limit?

In a proof of the completeness of $l^\infty$ on this page, the author finds a candidate limit point $x$ for a cauchy seqeuence $x^n$. He then chooses an $\varepsilon$ and then considers $\varepsilon/2$...
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1answer
118 views

Show the complex sequence converges to $\sqrt{2}$

Let $f(z) = \frac{z+2}{z+1}$ for $z\neq-1$. If $z_1=i$ and $z_{n+1}=f(z_n)$ for every $n\in\mathbb{N}$, then show that the sequence $(z_n)^\infty_{n=1}$ converges to $\sqrt{2}$. I was given the hint ...
4
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2answers
910 views

Is $C[0,1]$ not complete with the $L^2$ distance?

$$X=C[0,1]\qquad d(f,g)=\left(\int_{0}^{1}\vert f-g \vert^{2}dx\right)^{1/2}$$ The complete metric space is defined that every Cauchy sequence should be convergent on my book, so I think that the main ...
4
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1answer
74 views

Prove $\{g(x_n)\}_{n=1}^\infty$ converges

Let $g : (a, b) → R$ be uniformly continuous on $(a, b)$. Let $\{x_n\}_{n=1}^\infty$ be a sequence in $(a, b)$ converging to $a$. Prove that $\{g(x_n)\}_{n=1}^\infty$ converges. The general idea here ...
4
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3answers
71 views

Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
4
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2answers
35 views

How do I show that $s_n = id_E + (id_E-f)+ \dots + (id_E -f)^n$ is continuous and Cauchy sequence?

Let $E$ be a a Banach space. Let $f \in (\mathcal{L}(E), |||\cdot |||)$ such that $|||\text{id}_E-f|||<1$. We put $$s_n = id_E + (id_E-f)+ (id_E-f)^2 \dots + (id_E -f)^n$$ Show that $f$ is ...
4
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1answer
83 views

How do I prove this series diverges?

Consider a decreasing sequence $(x_n)$ in $\Bbb{R}_0$. There are an infinite amount of $n \in \Bbb{N}_0$ for which $1/n < x_n$. Prove the series $\sum x_n$ diverges. On one hand, I considered ...
4
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2answers
1k views

Every convergent sequence is a Cauchy sequence.

Today, my teacher proved to our class that every convergent sequence is a Cauchy sequence and said that the opposite is not true, i.e. Not every Cauchy sequence is a convergent sequence. However he ...
4
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1answer
319 views

Is $L^1_{loc}(\mathbb{R})$ complete with the norm $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy$

Let $BL^1_{loc}$ be the space of locally integrable functions $f:\mathbb{R}\to \mathbb{R}$ such that $|f|=\sup_{x\in \mathbb{R}}\int_x^{x+1}|f(y)|dy<\infty$. Is this space complete ? What I tried: ...
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1answer
60 views

Let $(a_n)$ and $(b_n)$ be Cauchy sequences of rationals. Then $(a_nb_n)$ is Cauchy sequence

Let $(a_n)$ and $(b_n)$ be Cauchy sequences of rationals. Then $(a_nb_n)$ is a Cauchy sequence. Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank ...
4
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1answer
74 views

Confusion about proof of Unit Balls are not compact in Infinite Dimensional Normed Spaces with Riesz's Lemma

We can construct a sequence such that $\|x_n-x_m\|\gt 1/2$ via using Riesz's Lemma. It's not Cauchy sequence and thus it's not a convergent sequence. My question : In my notes "since the sequence is ...
4
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4answers
124 views

How do I prove that $a_n=\sum_{k=1}^{n} \frac{1}{k}$ is not a Cauchy sequence?

First, I have to say that it does not make sense to me, because I know for sure that $\sum_{k=1}^{n}\frac{1}{k^2}$ is a Cauchy sequence so naturally in my mind the discussed one "should" converge too. ...
4
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1answer
299 views

How to show that $BV[0,1]$, the space of all functions on $[0,1]$ of bounded variation, is not complete under the supremum norm?

How to show that $BV[0,1]$, the set of all functions of bounded variation, is not complete under the supremum norm? Can one explicitly construct a Cauchy sequence which does not converge or find a ...
4
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3answers
184 views

Show that the sequence $a_n=1+\frac1{\sqrt{2}} +\frac1{\sqrt{3}} + \dots + \frac1{\sqrt{n}}$ is not Cauchy

Show directly (from the definition) that if $$a_n=1+\frac{1}{\sqrt{2}} +\frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}}\;,$$ then $(a_n)$ is not a Cauchy sequence. Attempt Using $a_n$ is ...
4
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1answer
875 views

Closed Sets VS. Complete Sets

Let $(X,d)$ be a metric space. If $K⊆X$, and $K$ is a closed set. Does that mean any Cauchy sequence in $K$ converges in $K$? If no, could someone give an example? If yes, then what is the ...
4
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2answers
184 views

Optimization of parameter for recursive Cauchy sequence

I have the following recursive sequence I'm analyzing: $$V_0 = 50, V_1 = (1-10k)V_0,$$ $$V_{n+1} = (1-10k)V_n - 5kV_{n-1}$$ where $k > 0$ is a parameter that I'm investigating by running ...
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4answers
2k views

Proving the usual distance metric in $\mathbb{R}$ is complete

If we allow the metric to be $d(x,y)=|x-y|$, we must prove that this is complete. Now, I have proven all properties of a metric space. However, I don't particularly now where to begin to prove that ...
4
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1answer
150 views

Quasi Cauchy sequences in general topology?

Suppose $(X,\tau)$ is a topological space and that $(X^2,\tau_2)$ is the product space. Now define $\mathscr S\!_\tau=\{W\in\tau_2|\Delta X^2\subseteq W\}$, where $\Delta X^2=\{(x,y)\in X^2|x=y\}$, ...
4
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1answer
116 views

Determine whether $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1,\infty)$

Detemine whether $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges uniformly on $(1,\infty)$. My attempt: Upon attempting to use the Weierstrauss M-test I get $$0\leqslant\|f_n(x)\|_\infty=\sup_{x\...
4
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1answer
71 views

Discover whether $\sum_{n = 2}^{\infty} \frac{1}{n\log(n)}$ is convergent or not, using Cauchy

I was asked if $\sum\limits_{n=2}^{\infty} \frac{1}{n\log(n)}$ was convergent or not. I already solved this problem using the integral property, but I wanted to use Cauchy instead. I defined $m,n \...
4
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1answer
96 views

A Cauchy sequence $\{x_n\}$ with infinitely many $n$ such that $x_n = c$.

Is the following argument correct? Proposition. If $\{x_n\}$ is Cauchy sequence such that $x_n = c$ for infinitely many $n$, then $\lim_{n\to\infty}x_n = c$. Proof. Let $\epsilon>0$. Since $\{x_n\...
4
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3answers
84 views

Prove directly from the definition that $({1\over2}+\frac{1}{2^2}+…+\frac{1}{2^n})_n $ is Cauchy

Prove directly from the definition that $({1\over2}+\frac{1}{2^2}+...+\frac{1}{2^n})_n$ is cauchy I know from the definition of Cauchy that |$x_n$-$x_m$|< ϵ but how do you do this with |$\frac{1}{2^...
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1answer
153 views

General approach to determine completeness of metric space

I've looked at a few questions online asking to determine the completeness of Metric Spaces. 2 such examples of metric spaces $(M,d)$: 1) $M = \{ (x,y) \in \mathbb{R}^2 \space : y>0 $ or $ x=0=y \...
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1answer
53 views

Is this proof involving complete metric spaces correct?

Show that if every closed ball of a metric space $(X, d)$ is complete then $ X$ is complete. I thought the following: given $(x_n)$ a Cauchy sequence in $X$, we have that the set $A= \{x_{1}, x_{2},.....
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1answer
215 views

How to decide completeness of $\ell^\infty$?

Let $\ell^\infty$ denote the set of all bounded sequences $x \colon = (\xi_j)_{j=1}^\infty$, $y \colon= (\eta_j)_{j=1}^\infty$ of complex numbers with the metric $d$ defined as follows: $$ d(x,y) \...
4
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0answers
295 views

Using Cauchy's Criterion to show non-uniform convergence of series of functions.

I want to show $$\sum_{n=0}^\infty x^n$$$$x\in(-1,1)$$ does not converge uniformly using the negation of Cauchy's Criterion for uniform convergence of series of functions. Cauchy's Criterion states ...