Proving the Liouville Numbers are uncountable
The definition I'm given for the Liouville numbers is just that $x$ is Liouville if for every $N > 0$ there exists integers $p,q \geq 2$ such that$$\left|x-\frac{p}{q}\right| < \frac{1}{q^N}.$$I...
View ArticleLebesgue-Measure on Euclidean Space and Natural way to extend Lebesgue...
I will ask two questions that involve proposition 1.15 and theorem 1.16 from Follands' real analysis book.First question:Folland constructs the Lebesbgue Stieltjes measure in $\mathbb{R}$, and I...
View ArticleUniformly convergence of power series
I know when r is radius of convergence of power seriesPower series is uniformly converge on Any closed interval of (-r , r)I want know when power series converges on (-r , r] , [-r,r] or [-r,r) each...
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Landau’s Analysis and Error in Peano’s definition of addition
I am working through Landau’s “Foundations of Analysis” and have been stopped in my tracks as early as theorem 4 (attached below, for reference). I spent a few days attempting to formulate my own...
View ArticleContinuity without topology nor $\epsilon$-$\delta$
I was reading this question, and the related answers, which popped out in merit to my previous question about continuity while searching over here: $\lim_{x\to 0} f(x)$ where $0$ is isolated in the...
View ArticleVerifying the integration Domain
I have a $n-2$ dimensional integral which I need to express in a more compatible form by changing variables. It'd be great if someone could verify the new domain of integration for me.Define, for...
View ArticleDoes this sequence of polynomials converge to the square root function?
Taken from Lang's R & F Analysis (p.60). For some reason I can't see why, for $t \in [0,1]$, the following is true for all natural numbers $n$ (by an inductive argument):$$0 \leq \sqrt{t} - P_n(t)...
View ArticleProve that an "interval" in $\mathbb{R}^n$ is an open set.
$\DeclareMathOperator{\interior}{Int}$I'm studying Real Analysis in $\mathbb{R}^n$ and came across the following exercise. Let $a=(a_1,...,a_n)$ and $b=(b_1,...,b_n) \in \mathbb{R}^n.$ Let us suppose...
View ArticleBounding a summation of positive terms
Bound the summation of the form$ \sum_{k=\tau}^{t} (1+k)^{-v} $where $ \tau $ and t are positive integers and v $ \in $ (0.5, 1].This summation is bounded by the form $ (1 + t - \tau)^{-v} / (1 -v) $I...
View ArticleApplications of Kolmogorov Arnold representation theorem (besides KANs)
A student of mine has recently asked me a question about the possible applications of Kołmogorov-Arnold Representation Theorem. While I am aware of the extent this theorem is utilized in the...
View ArticleTopology on $\mathbb{R}$ Induced by Lebesgue Density: Regular but Not Normal?
QuestionLet a neighborhood basis of a point $x$ of the real line consist of all Lebesgue-measurable sets containing $x$ whose density at $x$ equals $1$. Show that this requirement defines a topology...
View ArticleHelp with Infimum and Supremum in inequality.
I have a problem let $s_2 = \{x \in \mathbb{R} : x > 0\}$. Does $s_2$ have lower bound, upper bound? Does $\inf(s_2)$ and $\sup(s_2)$ exist?I understand the that the lower bound is 0 while there is...
View ArticleProb. 26, Chap. 5, in Baby Rudin: If $\left| f^\prime(x) \right| \leq A...
Here is Prob. 26, Chap. 5, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $f$ is differentiable on $[a, b]$, $f(a) = 0$, and there is a real number $A$ such that...
View ArticleGiven the function $f(x) = e^{-x^2}$, how can I find the $n$-th derivative of...
Given the function $f(x) = e^{-x^2}$, how can I find $f^{(n)}(0)$?The Taylor series for $e^{-x^2}$ can be written by expanding $e^u$ where $u = -x^2$:$$e^{-x^2} = \sum_{n=0}^{\infty}...
View ArticleThe double left continuous inverse is equal to the left continuous version of...
I have some non-decreasing function $f$, and I will denote the left continuous inverse of $f$ as $f^{\leftarrow}(y)=inf\{x:f(x)\geq y\}$. Now I want to...
View Articlecontinuous extension and smooth extension of a function
Let $X$ be a metric space. Let $E$ be a subset of $X$.(1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such that...
View ArticleMaximize $\sum x_{k}^{2}\sum kx_{k} $ for $x_1 \ge \cdots \ge x_n \ge 0$ and...
Let $ x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$, and $\sum\limits_{k=1}^{n}x_{k}=1$. What is the maximal value of$$\sum_{k=1}^{n}x_{k}^{2}\sum_{k=1}^{n}kx_{k} ?$$I think, Maybe we could try Rearrangement...
View ArticleSobolev spaces and ridge approximations
I am new to approximation theory and do not know much about Sobolev spaces.I aim to approximate a Sobolev function $f$ using linear combinations of ridge functions of the form...
View ArticleProve that $x_1^2+\cdots+x_6^2 \le 22$ subject to some constraints
Six varibles inequality:Given $x_1, x_2, x_3, x_4, x_5, x_6$ be non-negative real numbers satisfy \begin{cases} x_1 \ge x_2 \ge x_3 \ge x_4 \ge x_5 \ge x_6 \ge 0, \\ x_1-x_5 \le 2\sqrt{x_4\cdot x_6},\\...
View ArticleIf $u>0$ is $C^0(\Omega)\cap H^1(\Omega)$ and vanishes on the boundary of...
I'm trying to show that if $u$ is positive, continuous and $\Delta u\geq 0$ in $\Omega$, and if $u=0$ on $\partial \Omega$, then let $\tilde u$ be the zero extension of $u$, I want to say that $\tilde...
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