Questions tagged [cauchy-sequences]

For questions relating to the properties of Cauchy sequences.

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Homeomorphism between two complete spaces.

Let $X,Y$ be complete metric spaces and let $h: X \to Y$ be homeomorphism. If $(x_n)$ is a Cauchy sequence in $X$, is it true that $(h(x_n))$ is Cauchy in $Y$? Instinctively, I would go for no, ...
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Determine whether of the following sequences is cauchy and find the limits [closed]

Determine whether of the following sequences is cauchy and find the limits: $x_{1}$=1, $x_{n+1}$=$x_{n}$+1/n+1.
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Is the function space $X=\{u \in H^1(\Omega) : \text{$u$is continuous at$0$}\}$ complete?

This question is somewhat similar to Form functions that are continuous at one point in L^\infty a Banach space. where the continuity at zero was added to $L^\infty(\Omega)$. This space was indeed ...
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Why a Cauchy sequence in $W^{1, p}$ is also a Cauchy sequence in $L^p$

$W^{1,p}(I)= \{ u\in L^{p}(I) | \exists g \in L^p(I) :\int_{I}^{}u\varphi'= - \int_{I}^{}g \varphi \}$, where I = (a, b) $\subset \mathbb{R}$ with the norm $||u||_{W^{1,p}}=||u||_{L^p}+||u'||_{L^p}$ ...
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convergence of the series $\sum_{n=1}^{\infty}(1/n^{\alpha})(\int_{0}^{\frac{\pi}{4}}\tan^{n}tdt)x^{n}$

We are given $$\displaystyle W_n =\int_{0}^{\frac{\pi}{4}}\tan^{n}tdt.$$ Find the radius of convergence of the power series $$\displaystyle\sum_{n=1}^{\infty}\frac{W_n}{n^{\alpha}}x^n$$ in terms of ...
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Example of a sequence on an infinite-dimensional vector space with respect to different norms [closed]

I am stuck with the following problem: Give example of an infinite dimensional vector space V and two norms $\theta$ and $\rho$ on V and the sequence $\{x_{n}\}_{n\geq 1}$ of V such that: The ...
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Prove that the sequence $(a_n)$ is Cauchy and find the limit.

Let us define a sequence $(a_n)$ as follows: $$a_1 = 1, a_2 = 2 \text{ and } a_{n} = \frac14 a_{n-2} + \frac34 a_{n-1}$$ Prove that the sequence $(a_n)$ is Cauchy and find the limit. I have proved ...
$\left\{\frac{1}{n}\right\}$ converges to $\frac{1}{2}$ in some metric space $(\mathrm{X}, \mathrm{d})$
Consider the space $X=[0,1]$ Then A) $\left\{\frac{1}{n}\right\}$ converges to $\frac{1}{2}$ in some metric space $(\mathrm{X}, \mathrm{d})$ B) $\left\{\frac{1}{n}\right\}$ converges to 1 in some ...