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Prove that if $\{a_{k}\}$ is a sequence of real numbers such that...

Prove that if $\{a_{k}\}$ is a sequence of real numbers such that$$\sum_{k=1}^{\infty} \frac{|a_{k}|}{k} = \infty$$and$$\sum_{n=1}^{\infty} \left( \sum_{k=2^n-1}^{2^n-1} k(a_k - a_{k+1})^2...

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Prove that the union of increasing $\sigma$-field is not a $\sigma$-field

Find examples where $\mathcal{F}_1 \subseteq \mathcal{F}_2 \subseteq \cdots$ are increasing $\sigma$-fields but $\bigcup_{n = 1} ^\infty \mathcal{F}_n$ is not. I have constructed an example that I...

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How to proceed if $f$ is not conservative but almost is

I have been working in this problem about line integrals.Compute $\int_C \mathbf{f} d\mathbf{r}$ where $\mathbf{f}(x,y,z)=(2x,\cos(y)\cos(z),\sin(y)sin(z))$ and $C$ is the polygonal which links...

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Show that V is complete if and only if S is complete. [closed]

Let V be a normed space, and S be the unit sphere in V:S = {x ∈ V : ‖x‖ = 1}Show that V is complete if and only if S is complete.

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There's no injective, continuous function...

Assume a continuous and injective $$g:(\mathbb{Q},\lvert\cdot\rvert)\rightarrow(\mathbb{N},\lvert\cdot\rvert)$$Then $$\forall A\subset\mathbb{N}\implies\overline{f^{-1}(A)}\subset\ f^{-1}(A)$$ (because...

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How to complete this proof of $\int_{0}^ \infty \frac{nx...

I saw this problem:$\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx =\frac{\pi^2}{4}$ and I tried to solve it.Here is my attempt$\textbf{Claim: }$ If $f$ is continuous at $[0,1]$ then...

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Integral involve special functions [closed]

I was trying to evaluate a partial sum formula and I was stuck hereCan anybody help how to solve this integral$$\int_0^x \frac{t^{k + 1} - 1}{t - 1}dt $$

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Folland Real Analysis Proposition 1.3

The following is a proposition and its proof from chapter 1 of Folland's Real Analysis.1.3 Proposition. If $A$ is countable, then $\bigotimes_{\alpha\in A}\mathcal M_\alpha$ is the $\sigma$-algebra...

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Existence of a sequence of disjoint measurable sets such that the measures...

Let $(S,\Sigma ,\mu)$ be $\sigma $ finite measure space and $\mu$ does not concentrate on any finite sets.(For example, Lebesgue measure space.)I want to prove the existence of $(A_n)_{n\in...

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Does every bounded total ordered set have a supremum/infimum?

My question is really simple. I know intuitively every bounded total ordered set has supremum and infimum but I don`t know how to prove it formally. Must the set be complete?

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Bounding a sequence involving minimizers of strictly convex quadratic functions

Suppose that $H\succ 0$ and we have a sequence $\{x_j\}_{j\ge 1}$ such that they are the unique solution of the following problems:$$x_{j} \in \text{argmin}_{x\in \mathbb R^n}(x-x_{j-1})^T\nabla...

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Suppose f is a one-one function. Show that $f^{-1} \circ f(x) = x$ for all $x...

I am not sure if my answer below is complete, my math writing is pretty bad so I am not sure if I am covering all the necessary points. Please tell me if you think that the format/style is bad...

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Another non-trival Gaussian integral to solve

Does anyone know, how to solve the following integral:$$\int_{-\infty}^{\infty} e^{ax}\varphi(x)\Phi(bx)T(x,b)\ \mathrm{d}x,$$where $\varphi(x):={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}$ and...

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Separability of Lp spaces

I should show the following statements:$L^\infty(\mathbb{R}^n, \delta_{x_0})$ is separable;$L^p(\mathbb{R}^n, \#)$ is not separable for any p, where $\#$ is the counting measure on $\mathbb{R}^n$. Can...

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PDF of the difference of two Beta Prime distributions

I am struggling to find the PDF of the difference of two Beta Prime distribution.DefinitionA random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha,...

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Banach indicatrix

Good afternoon, I read Nathanson’s book, there is a theorem on the measurability of the Banach indicatrix, could you tell me in which theorems, proofs or problems the Banach indicatrix is ​​used in...

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Does the contraction mapping theorem ensure the uniqueness of the solution to...

In "Introduction to Smooth Manifolds" (IntSM) by J. M. Lee and "Ordinary Differenial Equations and Dynamical Systems" (ODEDS) by Thomas C. Sideris there is a separate proof for the uniqueness of the...

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Confusing different definitions of Lebesgue Integral of Simple Functions

Formally, a simple function is a finite linear combination of indicator functions of measurable sets. More precisely, let $(X, Σ)$ be a measurable space. Let $A_1, ..., A_n \in Σ$ be a sequence of...

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Bounding the series $\sum \frac{u_i u_j}{j-1}$

I was playing with some series, and came up with the following problem (which I am not sure is true):Let $(u_i)$ be a sequence of positive reals. Furthermore, suppose that $\sum u_i ^2 $ converges. Is...

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Bigger space than distribution space $\mathcal{D}(\Omega)$?

Is there some reasonable and nice space which contains the space of distributions and is strictly bigger?There are some easy measureable functions like $\mathbb{R}\to \mathbb{R}, x \mapsto...

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