The interval $(-\infty,\infty)$ is open/closed
Working in the reals, not the extended reals, I recently encountered several mathematicians, including my lecturer whose area of interest is topology, independently refer to $(-\infty,\infty),$...
View ArticleProving that the closure of a set is closed directly
Currently working through Rudin's principle's of mathematical analysis. I am trying to prove directly that the closure of a set is closed but am hitting a wall on one part of the proof. Namely, if we...
View ArticleExistence of absolutely convergent subseries given the base sequence...
My Real Analysis final is coming up and I'd like to practice working with sequences and series, so I picked a practice problem and tried working it out. The statement is the following:Let $ (x_n)_n $...
View ArticleCases where Simpson's rule has a greater error than Trapezoidal rule?
The error bound formulas for trapezoidal rule and simpson's rule say that:$\begin{array}{l}{\text { Error Bound for the Trapezoid Rule: Suppose that }\left|f^{\prime \prime}(x)\right| \leq k \text {...
View ArticleThe unit disc contains finitely many disjoint dyadic square whose area is...
This is Exercise 1.25.b in Pugh’s Real Mathematical Analysis.In this post, I showed that the unit disc contains finitely many dyadic squares whose total area is arbitrarily close to the area of the...
View ArticleDefinite integral involving K Bessel function and a square root
I have recently been trying to evaluate some integrals involving the modified Bessel function $K_0(x)$. The specific integrals are$$L(x,u) = \int_0^{1} K_0\left( 2x \sqrt{r(1-r)} \right) \exp(2ixur) dx...
View ArticleIs the set of probability measures on $[0,1]$ that induces a continuous...
I am an student in economics and I am trying to solve a fixed point problem, where the inputs of the function are probability measures. I'm trying to figure out whether the below results are true, but...
View ArticleDeriving a high order Derivative formula for fractional variable
Let $f\in C^{(\infty)}(\Bbb{R})$. Show that for $x\ne 0$:$$\frac{1}{x^{n+1}}f^{(n)}\left(\frac 1x\right)=(-1)^n\frac{d^n}{dx^n}\left(x^{n-1}f\left(\frac 1x\right)\right)$$I think a possible approach to...
View ArticleChange of Integral measure Involving a Supremum and Minimization with a...
Let $(X,\mathcal B, \mu)$ be a Standard Probability space and $0<\beta<1$. Let $A\in \mathcal B$ such that $0<\mu(A)<\infty$. Let $\varphi:A\to \mathbb R$ be a map. Define...
View ArticleIs limit always continuous extension in general topological space?
Let $X, Y$ be topological spaces, $D \subseteq X$, and $f: D \to Y$ a continuous function. Suppose that $a \in X \setminus D$ is a limit point of $D$, and there is some $y \in Y$ such that $f(x) \to y$...
View ArticleA Point in Sec. 7.8 in Apostol's "Mathematical Analysis ...", 2nd edition: Is...
This post is regarding a point made toward the end of Sec. 7.8 in the book Mathematical Analysis - A Modern Approach To Advanced Calculus by Tom M. Apostol, 2nd edition:While trying to bring out the...
View ArticleEffective lower bound on square root
Consider $r \in (0,1)$ and $t_1 <0$ and $t_2 \in \mathbb R.$I want to show that there exists a constant $C_r>0$ such that$$t_1 (1 - 2 r) + \sqrt{t_1^2 + 4 t_2^2 (1 - r) r} \ge C_r (\vert t_1\vert...
View ArticleThe convergence of unordered sum
$X$ is an arbitrary normed vector space. $A=\{x_i\in X\mid i\in J\}$ is an indexed set. $J$ contains the indices and is uncountably infinite. Let $\mathcal F=\{F\mid F\subseteq J, F \text{ is...
View ArticleIs this proof of the absolute convergence test for double sums correct?
Theorem:$$(\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} |a_{ij}| \space \text{converges}) \implies (\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} a_{ij} \space \text{converges})$$Proof:Let $\sum_{i=1}^{\infty}...
View ArticleSeeking feedback on my $\epsilon$-$\delta$ proof of $\lim_{x \to a} x^2 = a^2$.
I'm seeking feedback on my understanding of the $\epsilon$-$\delta$ limit proof for quadratic functions, specifically for $\lim_{x \to a} x^2 = a^2$.After studying multiple proofs, I've noticed that...
View ArticleTrying to prove monotone-sequences property ⇒ Archimedean property
Monotone-sequences property ⇒ Archimedean propertyToday I just started learning this and having trouble understanding parts of the proof. Sorry if these are easy to some, I just couldn't find this...
View ArticleProve that the function $\sqrt x$ is uniformly continuous on $\{x\in...
Prove that the function $\sqrt x$ is uniformly continuous on $\{x\in \mathbb{R} | x \ge 0\}$.To show uniformly continuity I must show for a given $\epsilon > 0$ there exists a $\delta>0$ such...
View ArticleCan a function be strongly differentiable but not continuously differentiable?
A similar question was asked before (however, there were a few issues with the definitions and answer given, as I pointed out over there): Can a function be differentiable but not strongly...
View ArticleAn idea for this difficult integral:...
I am being stuck in caculating this integral: $$J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}dx$$ I tried to change to another variable: $x = - t$ then $dx = - dt$,...
View ArticleIs the following method for proving density of irrational numbers in real...
The motivation for this question is:I told my friend to use:$\forall x_{1}, x_{2} \in \mathbb{R}, x_{1} < x_{2}, \exists r \in \mathbb{Q}: x_{1} < r <x_{2}.$To prove:$\forall x_{1}, x_{2} \in...
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