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...
View ArticleFor 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...
View ArticleSimple 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...
View ArticleInverse 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$$|\phi(T_n)...
View ArticleQuestion regarding the Generalized Mean Value Theorem
Suppose we are given functions $f$ and $g$ such that $f(x)\ge 0, g(x)\ge 0$$\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...
View ArticleAbout 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...
View ArticleProve 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...
View ArticleDoes 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$$\left( \lim_{t \to \infty} \vec{x}(t) = \vec{L} \quad \text{and} \quad \lim_{t...
View ArticleReal Analysis, Folland Theorem 1.18 Borel measures on the real line
Background information - We fix a complete Lebesgue-Stiltjes measure $\mu$ on $\mathbb{R}$ associated to the increasing right continuous function $F$, and we denote by $M_{\mu}$, the domain of...
View ArticleProve $ \frac{x_1-x_2}{x_n+x_1} + \frac{x_2-x_3}{x_1+x_2}+\cdots+...
Is it true that:$$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$$for $x_1,\ldots,x_n>0$, such that $x_1+\cdots+x_n=1$?It is a cyclic...
View ArticleA problem about asymptotics behaviour of the sum
Let $A\subset N$ be a set of zero density and let $a_1$ be its smallestelement. Also, let $L:[a_1, +\infty)\rightarrow\mathbb R_+$ be an increasing function which is continuous and differentiable on...
View Article$A_2 = {p \in \mathbb Q : p \leq 0} \cup {q \in \mathbb Q : q^2 \leq 2}$. Is...
So, for the set $A_2 = \{p \in \mathbb Q : p \leq 0\} \cup \{q \in \mathbb Q : q^2 \leq 2\}$ I know that the set is open, logically because $\sqrt2$ is not in the rationals, and because of that it's...
View ArticleCould someone explain "The subsets of Polynomials to degree n is not a...
The explanation my notes gives is that "This is because the sum of two polynomials of equal degree can have a lesser degree".I am not seeing how this is valid, if you add for example $$ P_1= 2x^3 -5x...
View ArticleRange 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...
View ArticleIs 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?$$\bigcap_{\delta>0}\{\|x\|:|f(x)|>1-\delta, |f(x)|\leq...
View ArticleProof explanation of Lemma 5.3.14 from Tao's Analysis I (3rd edition)
I do not understand parts in bold in the proof of the lemma which follows. Help me understand them.Lemma 5.3.14. Let $x$ be a non-zero real number. Then $x=$$\operatorname{LIM}_{n \rightarrow \infty}...
View ArticleProving 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$$v(A)=\sup\bigg\{\sum_{a\in F} a: F \text{ is a finite subset of }...
View ArticleNew Definitions of Upper & Lower Integrals
Let $f\colon [a,b]\to \mathbb{R}$ be a bounded function. To be simple, only upper Darboux sum and integral will be discussed.The upper Darboux sum of $f$ on $[a,b]$ based on a partition $P =...
View ArticleA cutoff function which the constant of its derivatives is independent of the...
In Page 842 of Function Spaces and Partial Differential Equations written by Ali Taheri, the author claims that there is a cutoff function which the constant of its derivatives is independent of the...
View ArticleProving $\sum_{i=m+1}^\infty \frac{m!^{i/m}}{i!}
One of these days, the user @AspiringMat was trying to prove that, for any integer $m\ge 1$,$$\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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