Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

1,515 questions
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
1k views

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: ...
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 ...
94 views

58 views

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 $|r^n|... 2answers 40 views When does subsequence convergence imply convergence? I know that for a sequence$\{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$I can ... 3answers 57 views Cauchy Sequence in subset of a metric space True or False: If E is a subset of a metric space X, then any sequence of points of E that is a Cauchy sequence in X is a Cauchy sequence in E. I'm having difficulty understanding this language. What ... 1answer 40 views Limit of sequence$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$. My ... 1answer 44 views Divergence of limit of function Let$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)$does not exist. Show ... 1answer 144 views nested interval property is equivalent to cauchy convergence criterion I can prove that nested interval property implies Cauchy convergence just by building intervals that become smaller and smaller and the diameter dwindles down to zero, intersecting in only one point. ... 0answers 22 views Cauchy implies monotone + bounded [duplicate] In my whole problem, I was stuck in one direction, which is to show (i) implies (ii). (i) Every Cauchy sequence in$\mathbb{R}$converges to a limit in$\mathbb{R}$. (ii) Every monotone and ... 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{2}+\sqrt{1}}+\frac{1}{\sqrt{3}+\sqrt{2}}+ ... +\frac{1}{\sqrt[... 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\... 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 ... 0answers 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|... 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 ... 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} ... 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 ... 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 ... 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 ... 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, ... 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 \... 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 ... 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 ... 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, ... 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 ... 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 ... 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 ... 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})... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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}...
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 ...
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 ...