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Is $d(x,y) = \sqrt{|x-y|}$ a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$.Is this a metric on $\mathbb{R}$?It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality seems...

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For which sequence does this summation method converges fast to $0$?

For a bounded sequence $(a_k)_{k=1}^\infty$ define the sequence $(A_j)_{j=1}^\infty$ by$$A_j = \sum_{k=1}^\infty \frac{a_k}{k^2+j^4}.$$It is elementary to check that (I cheated and used...

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Simple proof of weak version of Markov brothers' inequality

For a polynomial $f$, write$$\|f\|=\sup_{-1\leq x\leq 1}|f(x)|.$$It is a classical result of A. Markov that $\|f'\|\leq (\deg f)^2\|f\|$.Question. Is there a simple proof of the weaker result that, for...

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Inverse Property of Consistency [closed]

Given : $\{T_n\}$ is consistent for $\theta$then $P_{\theta}[|T_n - \theta|>\epsilon]\rightarrow 0$ as $n \rightarrow \infty $By continuity theorem for continuous funtion $\phi()$,we can$$|\phi(T_n)...

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Question regarding the Generalized Mean Value Theorem

Suppose we are given functions $f$ and $g$ such that $f(x)\ge 0, g(x)\ge 0$$\forall x \in [a,b]$ and $f,g$ are differentiable on $(a,b)$. Suppose $f'(x)=u(x)v(x)$ and $g'(x)=u(x)w(x)$ such that...

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About acceleration of the convergence of a sequence over another one.

Assume $u_n\to u$ in $L^2(\Omega)$, that is $\|u_n-u\|_{L^2(\Omega)}\to 0$. Let $\phi_\delta=\delta^{-d} \phi(\frac{x}{\delta})$ be a nice molifier $\phi\in C_c^\infty{B_1(0)}$, $0\leq \phi\leq 1$ and...

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Prove that finite unions of compact sets are compact

The question is: let $F_1, ... F_n$ be compact subsets of X. Show that $\cup^{N}_{n=1} F_n$ is compact. know that a set $ F \subset X$ is compact if every open cover $\mathcal {G}$ of F contains a...

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Does an asymptotically stationary curve in $\mathbb{R}^n$ with vanishing...

Looking for a proof or a counterexample: for twice continuously differentiable curves $\vec{x}(t) \in \mathbb{R}^n$, does$$\left( \lim_{t \to \infty} \vec{x}(t) = \vec{L} \quad \text{and} \quad \lim_{t...

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Real Analysis, Folland Theorem 1.18 Borel measures on the real line

Background information - We fix a complete Lebesgue-Stiltjes measure $\mu$ on $\mathbb{R}$ associated to the increasing right continuous function $F$, and we denote by $M_{\mu}$, the domain of...

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Prove $ \frac{x_1-x_2}{x_n+x_1} + \frac{x_2-x_3}{x_1+x_2}+\cdots+...

Is it true that:$$f_n(x_1,\ldots,x_n)=\frac{x_1-x_2}{x_n+x_1} + \frac{x_2-x_3}{x_1+x_2}+\cdots+ \frac{x_n-x_1}{x_{n-1} +x_n}\le 0$$for $x_1,\ldots,x_n>0$, such that $x_1+\cdots+x_n=1$?It is a cyclic...

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A problem about asymptotics behaviour of the sum

Let $A\subset N$ be a set of zero density and let $a_1$ be its smallestelement. Also, let $L:[a_1, +\infty)\rightarrow\mathbb R_+$ be an increasing function which is continuous and differentiable on...

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$A_2 = {p \in \mathbb Q : p \leq 0} \cup {q \in \mathbb Q : q^2 \leq 2}$. Is...

So, for the set $A_2 = \{p \in \mathbb Q : p \leq 0\} \cup \{q \in \mathbb Q : q^2 \leq 2\}$ I know that the set is open, logically because $\sqrt2$ is not in the rationals, and because of that it's...

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Could someone explain "The subsets of Polynomials to degree n is not a...

The explanation my notes gives is that "This is because the sum of two polynomials of equal degree can have a lesser degree".I am not seeing how this is valid, if you add for example $$ P_1= 2x^3 -5x...

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Range of parameter $a$ and an inequality related to zeros of $f(x) = ax^2 - x...

Suppose $ f(x) = ax^2 - x - \ln x$ (with a parameter $a\in \mathbb{R}$) has two distinct positive real zero points $x_{1}$ and $x_{2}$,i.e. $f(x_{1}) = 0$ and $f(x_{2}) = 0$.$(1)$ Calculate the range...

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Is it right $\bigcap_{\delta>0}\{\|x\|:|f(x)|>1-\delta, |f(x)|\leq...

Let $X$ be a Banach space and let $f:X\to \mathbb{R}$ be a bounded linear functional. Then is the following equality holds?$$\bigcap_{\delta>0}\{\|x\|:|f(x)|>1-\delta, |f(x)|\leq...

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Proof explanation of Lemma 5.3.14 from Tao's Analysis I (3rd edition)

I do not understand parts in bold in the proof of the lemma which follows. Help me understand them.Lemma 5.3.14. Let $x$ be a non-zero real number. Then $x=$$\operatorname{LIM}_{n \rightarrow \infty}...

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Proving set function is countably additive

Let set $X=\{\frac{1}{2^n}: n\in \mathbb{N}\}$ where $a_n=\frac{1}{2^n}$ for all $n\in\mathbb{N}$.For any $A\subseteq X$, define$$v(A)=\sup\bigg\{\sum_{a\in F} a: F \text{ is a finite subset of }...

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New Definitions of Upper & Lower Integrals

Let $f\colon [a,b]\to \mathbb{R}$ be a bounded function. To be simple, only upper Darboux sum and integral will be discussed.The upper Darboux sum of $f$ on $[a,b]$ based on a partition $P =...

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A cutoff function which the constant of its derivatives is independent of the...

In Page 842 of Function Spaces and Partial Differential Equations written by Ali Taheri, the author claims that there is a cutoff function which the constant of its derivatives is independent of the...

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Proving $\sum_{i=m+1}^\infty \frac{m!^{i/m}}{i!}

One of these days, the user @AspiringMat was trying to prove that, for any integer $m\ge 1$,$$\sum_{i=m+1}^\infty \frac{m!^{i/m}}{i!}<1.$$and asked for help here on MSE. I've spent too much of my...

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