I tried to find the boundary of $A=\{x\in \mathbb{R}^n\:|\:x_n\geq 0 \}$...
The set A is defined by $A=\{x\in \mathbb{R}^n\:|\:x_n\geq 0 \}$.I first pick any $r\in\mathbb{R}^{+}$, after that i defined B as$$B=\{y\in\mathbb{R}^n \:\vert\: y_n=0 \}$$Then i obtained the ball...
View ArticleWhether it holds that ${\mathcal B}(\Omega)\bigcap \Omega_0={\mathcal...
Given a Probability space $(\Omega,{\mathcal B}(\Omega),P)$, where $\Omega$ is a Polish space and equipped with the distance$$d:\Omega\times \Omega \rightarrow [0,\infty).$$${\mathcal B}(\Omega)$ is...
View Article$\epsilon-\delta$ proof for a (false) limit
I would like to understand how to complete this proof, for I am unable to conclude. Also I wanted to know if what I wrote before is correct. I tried to follow an answer over here (this one: How to...
View ArticleHow to prove triangle inequality for $p$-norm?
If $\mathcal{M}=\{M_i : i\in I_n\}$ is a collection of metric spaces, each with metric $d_i$, we can make $M=\prod_{i\in I_n}M_i$ a metric space using the $p$-norm, we simply set $d : M\times M\to...
View ArticleReference Request-second order mean value theorem
I found a term " the mean value theorem of the second order" in Hardy's A COURSEOF PURE MATHEMATICS page285.I'd like to know if there is any book which proves the theorem in detail?
View ArticleA counterexample of weak type (1,1) maximal function
Let $ f \in L_{\mathrm{loc}}\left(\mathbb{R}^2\right) $,$$\mathcal{R} = \left\{R \subset \mathbb{R}^2 : R \text{ is a rectangle with sides parallel to the coordinate axes} \right\}.$$Define the strong...
View ArticleHelp to show relation between ${\sqrt {x + 1}} - {\sqrt x} \leq...
Show that ${\sqrt{x+1}} - {\sqrt x} \leq {\frac{1}{2\sqrt x}}$ for all $x \in (0, \infty)$.I know that the derivative of ${\sqrt x} + 1$ is equal to ${\frac{1}{2\sqrt x}}$. But I don't know if there is...
View ArticleSup Norm and Uniform Convergence
My book says Convergence in sup norm $||f_n-f||\to 0$ is equivalent to uniform convergence and this follows immediately from definitions. but I just want to check:$\Rightarrow$ If...
View ArticleUnsure if $\sum_{n=1}^{\infty} \frac{1}{(5 + \cos n)^n}$ converges or diverges
I am unsure if the series $$\sum_{n=1}^{\infty} \frac{1}{(5 + \cos n)^n}$$ converges or diverges.I know that $5+\cos n$ lies between $5-1^n=4$ and $5+1^n=6$, meaning that we can bound the expression as...
View ArticleIs a piecewise smooth simple closed curve the union of countable number of...
Regarding the question "Is a smooth simple closed curve the union of finitely many arcs?", I want to consider a more general case when we choose the graphs of only one type, wlog let it be...
View ArticleE $\subseteq$ R，n $\in$ N. $E_n = \{y\in R:|x-y|
I don't know whether $E= N $ is a set that satisfies the conditions. When $n \to \infty$, is $E_n=N$? Then, $\lim_{n \to \infty} m(E_n) = m(E) $
View ArticleTaylor expansion of distribution
There is a well-known delta function identity which allows for the expansion of$$\chi_\epsilon(x)\equiv\left(\frac{1}{\epsilon^2+x^2}\right)^a,\quad x\in \mathbb{R}^n$$see for example this Math.SE...
View ArticleMeaning of $d(x_m,x_n)\to 0$ on $\mathbb{R}$
Consider the next lemma: Lemma. A sequence $(x_n)_{n\in\mathbb{N}}$ on a metric space $(X,d)$ is a Cauchy sequences, if and only if, $d(x_m,x_n)\to 0$ on $\mathbb{R}$.I don't understand the formal...
View ArticleHow to prove periodicity of $\sin(x)$ or $\cos(x)$ starting from the Taylor...
With the Taylor series representation of $\sin$ or $\cos$ as a starting point (and assuming no other knowledge about those functions), how can one:a. prove they are periodic?b. find the value of the...
View ArticleProve that a continuous function maps bounded sets to bounded sets
Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a continuous function, and $S \subset \mathbb{R}^n$ a bounded set. Now I want to show that $f(S)$ is bounded.My thoughts: Since $S$ is bounded it's contained...
View ArticleMeasurable wrt to sum implies measurable wrt to summands
Assume that $(X, \mathcal{R}, \mu)$ and $(X, \mathcal{R}, \nu)$ are premeasures on a ring $\mathcal{R}$ such that there exists $\{A_n\} \subset \mathcal{R}$ where $A_n \uparrow X$ and $\mu(A_n) <...
View ArticleAn inequality including maximal function
Let $ f \in L^1\left(\mathbb{R}^n\right) $. Prove that for any $ \lambda > 0 $,$$\left|\left\{x \in \mathbb{R}^n : M f(x) > \lambda \right\}\right| \leq \frac{C}{\lambda} \int_{(|f| > \lambda...
View Article$\lim_{x \to 0}\sqrt{1-x^2}=1$ proof using epsilon-delta
To show $\lim_{x \to 0}\sqrt{1-x^2}=1$:For any $\epsilon>0$, $\exists\delta$: $|x-0|<\delta \implies \epsilon>|f(x)-1|$$|x|<\delta \implies...
View ArticleClik here to view.
Seber, Ex. 1.b.3) - Unconstrained optimization problem
In one of my previous questions, I tried to solve the following exercise:Now the solution to exercise 3a) mentions that one may alternatively substitute $w_n = \sum_{i=1} ^{n-1} w_i$ and turn the...
View ArticleA property of the space $W_{0}^{1,p}$ concerning $u\le v \in W^{1,p}(\Omega)$...
In Gilbarg-Trudinger Section 8.1. The Weak Maximum Principle, they define $u\le v \in W^{1,2}(\Omega)$ on $\partial \Omega$ by $(u - v)^+\in W_0^{1,2}(\Omega)$.My question is: does the following...
View Article