Prove for all sequences $\{a_n\}$ and $\{b_n\}$, if $\lim a_n = a$ and $\lim...
I already know how to prove this statement in "English," but I would to see a proof of it entirely in first-order logic. Here is the English proof:(1) Let $\{a_n\}$ and $\{b_n\}$ be arbitrary...
View ArticleInverse rule to the L'Hôpital's rule
If in L'Hôpital's rule we have that: $f,g :(a,b)\to \mathbb{R}$, there exist $f'(x)$, $g'(x)$, and $g'(x)\ne0$,$$\lim_{x\to a^+} \frac{f(x)}{g(x)} = L,$$and also $\lim_{x\to a^+} f(x)=\lim_{x\to a^+}...
View ArticleSummation of products of oscillatory sequence
I encounter a sum in the form\begin{equation}\sum_{j=1}^{\infty}a_{j}b_{j}e^{i j\theta}\end{equation}I expect a lot of cancellations to happen due to the oscillatory term $e^{ij\theta}$.I can control...
View ArticleProving that Monotone Convergence implies Least Upper Bound in $\mathbb{R}$.
I tried proving that Every bounded increasing sequence converges in $\mathbb{R}$.implies that $\mathbb{R}$ has the least upper bound. Here, $\mathbb{R}$ is taken as an ordered field which contains the...
View ArticleInterpretation of a sigma-algebra not generated by a random variable.
I am wondering what is the interpretation of a sigma-algebra that is not generated by a random variable?For example if we have a random variable $X$ on a probability space $(\Omega, \mathcal{A},P)$...
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Examples of *uncountable* null and meagre sets that are not rare
I'm trying to teach myself real analysis, and I'm trying to figure out the various notions of "small" sets. My current understanding lead me to the following Euler diagram:For the other regions in the...
View ArticleOn the continuity of a function, again
Sorry to bother you with another question on continuity, but I need experts to explain me things. I cannot interact with books only.So the question is this: we know a definition of continuity thought...
View ArticleGeometric Interpretation of the Jacobian Matrix and Its Eigenvectors
I understand that for scalar-valued functions $g: \mathbb{R}^n \to \mathbb{R}$, the gradient represents the direction of maximum ascent. Similarly, for vector-valued functions $f: \mathbb{R}^n \to...
View ArticleUsing the definition of the substitution rule to calculate the antiderivative...
My task is the following:Find all antiderivatives of $\cos(3x-1)$.I instantly thought that I have to do that with the substitution rule, but I don't understand the definition that is given and I am not...
View ArticleProve that if $E_{\alpha}^+=$ {$x \in R :f_+^*(x) \gt \alpha$} ,then...
We define the "one-side" maximal function $$f_+^*(x)=\sup_{0 \lt h}\frac{1}{h} \int_x^{x+h} \vert f(y) \vert \, dy$$ Prove that if $f$ is integrable and $E_{\alpha}^+=$ {$x \in R :f_+^*(x) \gt \alpha$}...
View ArticleNumerical stability motivation of using Log Sum Exp for Geometric Programming?
I am an engineer who is currently working with some network optimization problem. Recently, I have been learning about Geometric Programming (GP). It seems to me that there are 2 approach for solving...
View ArticleDetermine the covergence of the sequence [duplicate]
Determine whether the sequence defined by $a_n=n^2 \cos \left(\frac{2}{n^2}-\frac{\pi}{2}\right)$ converges or diverges. If it converges, find the limit.I know that this is convergent. But which value...
View ArticleIs the CDF of the Gaussian Distribution in $L_p$ Space for $x \leq 0$?
Consider the cumulative distribution function (CDF) of the Gaussian distribution, and denote it as $\Phi(x)$.For the whole real line $\mathbb{R}$, $\Phi(x)$ is not integrable, i.e.,$$\int_{x \in...
View ArticleIs this function locally constant almost everywhere? [closed]
Let $g:(0,1) \to \mathbb{R},\, x \mapsto g(x)$ be Lipschitz continuous. Define the function$$X : (0,1) \to \{0,1\},\, x \mapsto \left\lbrace \begin{array}[cc]1 1 & \text{, if }\, g(x) = 0 \\ 0...
View ArticleProving $r(t) \to 0$ for a spiral
This is a problem from Ted Shifrin's book on Differential Geometry; consider the spiral $\vec{\alpha} (t) = {r}(t)(\cos t, \sin t)$ where $r(t)$ is a $C^1$ function bounded between $[0,1]$....
View ArticleHausdorff metric and Vietoris topology
I am supposed to show that on a compact metric space, the Hausdorff metric and the Vietoris topology induce the same topology. Does anybody know how this can be done? I wanted to start by showing that...
View ArticleBauer Maximum Principle
I came accross the Bauer Maximum Principle:"Any function that is convex and continuous, and defined on a set that is convex and compact, attains its maximum at some extreme point of that set."Why does...
View ArticleShow that $\left \lVert \int_0^1 F(t) dt \right \rVert \leq \lVert F...
Let $F:[0,1] \to \mathbb{R}^n$ be a continuous function such that $F=(f_1,...,f_n)$ and let:$$\int_0^1 F(t) dt := \left( \int_0^1 f_1(t) dt,...,\int_0^1 f_n(t) dt\right)$$And:$$\lVert F \rVert_{\infty}...
View ArticleDetermining the Number of Possible Sign Combinations for $n$ Elements
I want to know all the possibilities for a combination of $n$ elements when taking their signs into consideration.If $n=1$, meaning that we consider the set $\{a_1\}$, there are $N=2$ options:$$a_1...
View ArticleThe proof about the dual space of $L^1$ and $L^{\infty}$
Let $(X,\mu,M)$ be a measure space with $X=[0,1]$, $\mu$=counting measure and $M$ is the $\sigma$-algebra contains all the subset $E$ of $[0,1]$ such that either $E$ or $E^c$(complement) is countable....
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