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Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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19 views

Is there a space where no convergent Cauchy sequences exist?

I understand every non-complete space can be made complete by including the limit points of convergence of Cauchy sequences. I assume that the opposite process is possible as well. But, do spaces ...
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2answers
66 views

Prove that $(X,\Vert {\cdot}\Vert_2)$ is not complete where $\Vert f\Vert_2=\left(\int_{-2}^{2}|f(t)|^2 dt\right)^{1/2}$ [duplicate]

Prove that $(X,\Vert {\cdot}\Vert_2)$ is not complete where $X=C[-2,2]$ and \begin{align}\Vert f\Vert_2=\left(\int_{-2}^{2}|f(t)|^2 dt\right)^{1/2}.\end{align} MY TRIAL It suffices to produce a ...
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3answers
127 views

Cauchy sequence and subsequence [closed]

"Let $\{x_n\}\subset U$ be a Cauchy sequence. Give a direct proof that if a subsequence $\{x_{nk}\}\subset S$ has a limit $L$, then the Cauchy sequence $\{x_n\}\subset U$ has $L$ as a limit. Do not ...
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3answers
62 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 ...
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2answers
48 views

Show that $|\frac{1}{2n}-\frac{1}{2m}| < \epsilon$ holds for all $m, n > \frac{1}{\epsilon}.$

In Example 1.5-9 of the book Functional Analysis by Kreyszig it claims that $|\frac{1}{2n}-\frac{1}{2m}| < \epsilon$ holds for all $m, n > \frac{1}{\epsilon}.$ My calculations don't lead to ...
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2answers
43 views

Prove a closed subset of a complete metric space is complete via contradiction

I've seen some proofs by definition and just want to ask for proof verification on whether this is okay as well: Given complete metric space $(X,d)$ and $A \subset X$, $A$ closed. Prove $A$ is ...
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1answer
67 views

Continuous linear operator preserving Cauchy sequence in metric vector spaces

Let $T: X \rightarrow Y$ be a continuous linear operator. $(X,\rho), (Y,\xi)$ are linear metric spaces and $\{x_n\} \subset X$ is a Cauchy sequence. I need to show that $\{Tx_n\}$ is a Cauchy ...
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6answers
1k views

Making sure if it is Cauchy

In my real analysis exam I had a problem in which I proved that $|x_{n+1} - x_n|\lt {a^n}$ for all natural numbers $n$ and for all positive number $a\lt 1$ then $(x_n)$ is a Cauchy sequence. This was ...
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1answer
34 views

Is $T$ bounded if $(X,d)$ is not complete?

I) Let $b_1,b_2,...,b_n....$ be a Cauchy sequence in a complete metric space $(X,d)$. If $T$ is the set of points in the sequence show that $T$ is a bounded set. As the space is complete, every ...
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1answer
37 views

Banach space with subset whose elements are at least $d\gt 0$ far from each other is not separable

Let $X$ be a Banach space, and $A\subseteq X$ subgroup, where $A$ is not countable, and there is some $d \gt 0$ such that for all $x,y \in A$: $||x-y||>d$. Prove that $X$ is not separable. My ...
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6answers
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A sequence in $\mathbb{R}$ that has no Cauchy subsequence

Give an example of a sequence in $\mathbb{R}$ which has no subsequence which is a Cauchy sequence. I can find out a sequence that is not a Cauchy sequence such as $\{\ln(n)\}$ once $|\ln(n)-\ln(n+1)|=...
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1answer
39 views

Proving if a sequence is Cauchy and/or convergent. [duplicate]

Let $M=(0,\infty)$ be supplied with the metric function $d(x,y)=|\arctan(x)-\arctan(y)|$ and let $\{n\}_{n=1}^\infty$ be a sequence of positive integers. a) Is the sequence a Cauchy sequence in $(M,d)...
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1answer
34 views

Any Cauchy net in metric space is convergent?

Any Cauchy net in a metric space is convergent? I know that if (X,d) is metric space COMPLETE then cauchy net in X is convergent in X. But, only (X,d) is metric space (not complete) It is true? pd: ...
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79 views

Square Root (Cauchy Sequence)

So I'm typing on a phone or else I'd use math Jax, but basically take an infinite sequence that is made of the values of the square root function as each successive term. This clearly does not ...
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2answers
94 views

Prove that if a sequence is unbounded, then the sequence is not Cauchy

So far: If a sequence is unbounded, therefore it is a monotonic, divergent, sequence. Choosing $\epsilon = \frac12$, and $N\in\mathbb{N}$. Assume n, m $\geq N$. Then $\left| a_n - a_{N} \right| < ...
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1answer
52 views

Show that (X,d) is complete

Suppose $(X,d)$ is a metric space and for all sequences $ \{x_n\} $, if $\sum d(x_n,x_{n+1})< \infty $ then $ \{x_n\} $ converges. Prove that $(X,d)$ is complete. Question: I can show that if $\...
4
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3answers
58 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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2answers
40 views

When does subsequence convergence imply convergence?

I know that for a sequence $\{x_n\}$, if $\{x_{2n}\}$ and $\{x_{2n+1}\}$ converge to the same limit, then $\{x_n\}$ converges and to the same limit. My question is if I know that from any $x_i$ I can ...
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3answers
57 views

Cauchy Sequence in subset of a metric space

True or False: If E is a subset of a metric space X, then any sequence of points of E that is a Cauchy sequence in X is a Cauchy sequence in E. I'm having difficulty understanding this language. What ...
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1answer
40 views

Limit of sequence $x_n$

For $\alpha \in (0,1)$ let the sequence $\{x_n\}$ be such that $x_0 = 0, x_1 = 1 $ and $x_{n+1} = \alpha x_n + (1-\alpha)x_{n-1},\quad n\geq1$. Find $\displaystyle \lim_{n\rightarrow \infty}x_n$. My ...
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1answer
44 views

Divergence of limit of function

Let $f$ be a bounded real-valued function on a subset of $\mathbb{R}$ and let $x_{0} \in \mathbb{R}$ be a cluster point with respect to $A$. Suppose that $\lim_{x\to x_0} f(x)$ does not exist. Show ...
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1answer
144 views

nested interval property is equivalent to cauchy convergence criterion

I can prove that nested interval property implies Cauchy convergence just by building intervals that become smaller and smaller and the diameter dwindles down to zero, intersecting in only one point. ...
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0answers
22 views

Cauchy implies monotone + bounded [duplicate]

In my whole problem, I was stuck in one direction, which is to show (i) implies (ii). (i) Every Cauchy sequence in $\mathbb{R}$ converges to a limit in $\mathbb{R}$. (ii) Every monotone and ...
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1answer
78 views

How to investigate the convegence of this sequence?

How to investigate the convegence of the following sequence using Cauchy's convergence test? $$x_{n} = \frac{1}{1}+ \frac{1}{\sqrt[1]{2}+\sqrt{1}}+\frac{1}{\sqrt[2]{3}+\sqrt{2}}+ ... +\frac{1}{\sqrt[...
4
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1answer
90 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\...
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3answers
81 views

Use Cauchy Criterion to prove that $1+\frac{2^2}{2!}+\frac{2^3}{3!}+\cdots+\frac{2^n}{n!}$ converges

I've found similar problems, but they all have ones in the numerator or something that could be eventually substituted with ones, so their solutions won't work. Here's my attempt using the following ...
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53 views

Convergence of an recursive sequence with Cauchy

Let's assume we've shown $|x_{n+1}-x_n| \leq q|x_n-x_{n-1}|$ converges and is cauchy for $0<q<1$ like in this prove here: Suppose that $(x_n)$ is a sequence satisfying $|x_{n+1}-x_n| \leq q|...
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1answer
39 views

Proof that in Сauchy metric space the union of a sequence of closed sets with no internal points has no internal points

Proof that in Сauchy metric space the union of a sequence of closed sets with no internal points has no internal points. My attempt: Let's proof the statement for two sets by the contradiction ...
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2answers
56 views

How to show that $C_0 (U) $ is a Banach space

Let $ \emptyset \neq U \subset \mathbb{R} ^d $ be an open set. Define $ C_0(U) := \left\{ f \in C(U) : \forall \epsilon > 0 \exists K \subset U , K \mbox{ compact and }\sup_{x \in U \setminus K} ...
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1answer
68 views

Show that $C^1[a,b]$ with the uniform norm is not a Banach space

Consider the normed vector space $(C^1[a,b],||\cdot||_{\infty})$, we have to prove that it is not Banach. From theory a space is Banach if it is complete, ie if every Cauchy sequence in the space ...
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1answer
153 views

Prove $(l^{\infty},d_{\infty})$ is complete (metric space)

I need to prove that the space of the limited sequences $l^{\infty}$, with the metric $d_{\infty}\left((x_{n}),(y_{n})\right)=\sup\{|x_{n}-y_{n}| : n\in\mathbb{N}\}$ is complete. I saw this question ...
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1answer
43 views

If (X,d) is a metric space and $K \subseteq X$ is compact. Show that $K$ is complete.

I've seen other proofs of this on here but all of them seem to rely on showing compact iff sequentially compact in a metric space but instead I want to use only the definition of compactness, that ...
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1answer
70 views

If f is a uniform continuous function on (a,b), then f is bounded on (a,b).

If $f$ is uniformly continuous on $(a, b)$, then $f$ is bounded on $(a, b)$. I'm trying to understand this proof given here in this link. Its the second proof given from the question in the link, ...
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0answers
60 views

Equivalent statement on Cauchy sequence

There are three statements that I need to show their equivalence: (i) Every Cauchy sequence in $\mathbb{R}$ converges to a limit in $\mathbb{R}$. (ii) Every monotone and bounded sequence in $\...
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2answers
55 views

Find a divergent sequence whose distance lessens with each subsequent term

The question states Give an example of a sequence $(a_n)$ such that $\lim_{n\to\infty} |a_{n+1} − a_n| = 0$ but which is divergent. I'm slamming my head against a table thinking of all the divergent ...
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5answers
107 views

$\sum_{n=1}^{\infty} \left\{ e-(1+\frac{1}{n})^n \right\}$. is this converge or diverge

$$\sum_{n=1}^{\infty} \left\{ e-\left(1+\frac{1}{n}\right)^n \right\}$$ Is this converge or diverge series .It is a series with positive terms ,but none of test of positive term series is seems to ...
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1answer
77 views

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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2answers
52 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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0answers
44 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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1answer
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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0answers
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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1answer
81 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
67 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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1answer
49 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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4answers
143 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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0answers
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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2answers
36 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 ...
0
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1answer
38 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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2answers
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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1answer
29 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 ...