How does distance work? [duplicate]
I'm reading a book about PDEs and it opens with a discussion of set theory and limit points and whatnot and this got me thinking about how we define distance (since it uses neighborhoods to talk about...
View ArticleProve that this sequence is eventually one
Let $ \ \mathbb{N} = \{ 0,1,2,3,4,...\} \, $, $ \ O = \{ n \in \mathbb{N} : n \text{ is odd} \} \ $ and $ \ T: O \to O \ $ be such that, for all $ \ n \in \mathbb{N} \, $,\begin{align*}T(8n+1) & =...
View ArticleAsymptotics of 2Ai(x)Ai'(x)
I'm looking for the asymptotics of the derivative of Airy squared $2Ai(x)Ai'(x)$ as $x\to \infty$. I know that $A i^2(x) \sim \frac{1}{4} \pi^{-1}(x)^{-1 / 2} e^{-\frac{4}{3}(x)^{3 / 2}}$. Would it be...
View ArticleUnderstanding ordered fields and the subset $P \subseteq \mathbb{F}$ of...
I'm following Real Analysis: A Long-Form Textbook (Jay Cummings) and there is a part about defining the positive set $P \subseteq \mathbb{F}$.The following definition is given:An ordered field is a...
View ArticleTrying to solve an exercise on double integration
I have a problem with the following integral:$$\iint_{\Omega}y^{2}\,{\rm d}x\,{\rm d}y,$$$$\mbox{where}\quad \Omega \equiv \left\{\left(x,y\right) \in \mathbb{R}^{2}\ \mid\x^{2} + y^{2} \leq...
View ArticleFind infimum of $\{\frac{1}{n^2} | n \in \Bbb N\}$ in $\mathbb{R}$
So I need to find the infimum of$\{\frac{1}{n^2} | n \in \Bbb N\}$I know that this means that I need to find some $x$ where $x < \frac{1}{n^2} \forall n \in \Bbb N$.By intuition I know that $\lim_{n...
View ArticleApproximate continuous functions by "partially injective" functions.
In Munkres book Topology, Theorem 50.5, page 311, the author gives the following definition in a proof.Let $f:X\to \Bbb R^n$ be a continuous function where $X$ is a compact metric space. Define...
View ArticleCoordinate-free definition of smooth map
Say $(\mathcal E, V)$ and $(\mathcal F, W)$ are Euclidean spaces. A map $F : \mathcal E \to \mathcal F$ is said to be differentiable at $p_0 \in \mathcal E$ if there exists a linear map $L : V \to W$...
View Article$\lim_{n\to+\infty}\int_{0}^{2\pi}\left(1+\frac{\sin(x)}{n}\right)^{\frac{n}{...
I've been trying to compute the following limit:$$\lim_{n\ \to\ \infty}\int_{0}^{2\pi}\left[1 + \frac{\sin\left(x\right)}{n}\right]^{n/x}\,{\rm d}x$$I am not completely sure, but when $x\in (0,2\pi)$...
View ArticleConvergence rate of $(1+x/n+\dots)^n$
It is a well known fact that$$\lim_n (1+x/n+O(n^{-3/2}))^n=e^x$$For example, this is a key step in the standard proof of the central limit theorem.What can we say about the rate of convergence of this...
View ArticleShow that the only tempered distributions which are harmonic are the the...
Let $d\geq 1$. Using the Fourier transform, show that the only tempered distribution $\lambda \in\mathcal{S}(\mathbb{R}^d)^*$ which are harmonic (by which we mean that $\Delta \lambda=0$ in the sense...
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Why does swapping values work? [closed]
Why does putting negation of $f_1$ and $f_2$ work, as it says below (21)?Can someone please explain?
View Articlebook recommendation for real analysis
During the next quarter at uni, I'll be taking a course in real analysis and since I prefer studying with an additional text I thought I'd come here to look for some book recommendations.My background:...
View ArticleA non-measurable set with density $\frac{1}{2}$
According to Lebesgue's Density Theorem:Let $\mu$ be the Lebesgue outer measure, and let $A\subseteq\mathbb{R}$ be a Lebesgue measurable set. Then the limit:...
View ArticleLagrange Multipliers with EVT
Just to confirm, if EVT applies on the constraint, i.e. there exists an absolute max and min, and applying Lagrange multipliers yields two possible points, then is it guaranteed that one of those...
View ArticleBaby Rudin Chapter 2, Exercise 2 proof check
I am currently self-studying Baby Rudin and I have written what I think is a solution to Exercise 2 of Chapter 2. It reads:$\textbf{Exercise 2}:$ A complex number $z$ is said to be algebraic if there...
View Articlecondition to apply integration by parts?
I have two real functions $f, g$ defined on $[a,b]$. Both functions are differentiable on the open interval $(a,b)$.Is this enough to apply integration by parts$$\int_{a}^b f(x) g(x) dx = f(b) G(b) -...
View ArticleNon-symmetric bump function
I am trying to create a smooth bump function $f(x;a,b,c,k_1,k_2)$ satisfying the following properties:$f$ is smooth (can be relaxed)$f$ is supported on $[a,c]$.$f$ is increasing on $(a,b)$.$f$ is...
View ArticleRelation between the second derivative and the second degree polynomial...
Let E $\subseteq \mathbb{R}$, $a$ be a limit point of $E$, $c \in \mathbb{R}$ and $f, h: E \to \mathbb{R}$ be differentiable functions such that $f(x) = f(a) + f'(a)(x - a) + c(x - a)^2 + h(x)$ for all...
View ArticleMeasure associated to locally integrable function is regular
Suppose $f \in L^{1}_\text{loc}(\mathbb{R}^{d})$ with $f \geq 0$ and create the measure $\mu$ such that $d\mu = f\, dx$ where $dx$ is the standard Lebesgue measure. Must $\mu$ be a regular...
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