Analysing a Lebesgue integral inequality for $|t^{-n} \phi(x/t)|$, where...
Context. Let $C^k(\mathbb R^n)$ denote the space of functions defined on $\mathbb R^n$ that are $k$ times continuously differentiable, where $k \geqslant 1$ is an integer. As usual, define...
View ArticleWhen $x$ tends to $3$ is $x+3$ always less than $6$ or not?
In a question, it was asked to prove that as limit $x$ goes to $3$, $x^{2}$ tends to $9.$Now, on solving it by the $\epsilon$-$\delta$ method, when I simplified $x^{2} - 9$ to...
View ArticleConvergent subsequences in $L^p(\mathbb R^n)$
This is a particular fact that I ended up proving in the process of attempting one of my recent homeworks, but I don't think I've seen this particular fact online even though it feels like a fairly...
View ArticleHow Cantor's diagonal argument could be applied to $f:\mathbb N \to \mathcal...
Consider $f:\mathbb N \to \mathcal P (\mathbb N)$ being a bijection from $\mathbb N$ to $\mathcal P (\mathbb N)$ such that every natural number is associated to a subset of $\mathbb N$ given by the...
View ArticleHow can we use analysis to determine the error in computing a...
What is the relative error $\delta = \frac{|\Delta f|}{|f|}$ in computing the value of a function $f(x,y,z)$ at a point $(x,y,z)$ whose coordinates have absolute errors $\Delta x, \Delta y, \Delta...
View ArticleConditions that a sequence should satisfy to be an eventually monotone sequence
Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that:$a_n\in[0,1]$, $\forall n\in\mathbb{N}$$\lim_{n\to\infty}a_n = 0$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = 1$$a_{n+1} \leq a_n$,...
View ArticleProve that the number $0.a_1a_2a_3\ldots$ is a rational number.
Let $a_1$ be any number from the set {$0, 1, 2, \ldots, 9$}. For each $n \in \mathbb{N}$, denote by $a_{n+1}$ the last digit of the number $19a_n + 98$ in decimal notation. Prove that the number...
View ArticleRemoving that $x_0$ must be an accumulation point from the definition of limit.
The textbook "Elementary Real Analysis" by Thomson, Bruckner suggests that the definition of $lim_{x\to x_0}f(x)$ requires that $x_0$ is an accumulation value of the domain of that function. One of the...
View ArticleExtracting a subsequence common to infintely many sets from an uncountabe...
Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning...
View ArticleInjective functions $f : \mathbb{R} \rightarrow\mathbb{R}$, such that for any...
Find all injective functions $f : \mathbb{R} \rightarrow \mathbb{R}$, such that for any $x, y \in \mathbb{R}$, they satisfy the condition: $f(xy) = f(f(x)f(y))$.Attempt: Because f is injective,...
View ArticleEvaluating $\lim_{n\to\infty}\sum_{i=1}^n...
Let $X_1,...,X_n\stackrel{iid}{\sim} \mu$ where has a density with respect to the Lebesgue measure on $\mathbb{R}$: $\mu(dx)=\rho(x)dx$. For every $x$ show$$\lim_{n\to\infty}\sum_{i=1}^n...
View ArticleMinimize the area of the convex hull of the roots of the integral of...
Recently I learned that the convex hull of the roots of a complex polynomial contains the roots of its derivatives, so I asked my self if you can minimize in the choose of the constant in an integral...
View ArticleMeasure on the $d$-dimensional torus
I am looking for references, measure-integration theory where the $d$-dimensional torus $\mathbb{T}^d$ is treared rigorously: borel $\sigma$-algebra, measure functions, measures on...
View ArticleProve that $|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|$.
The following problem is the final exercise of problem set 1 on MIT OCW's course 18.100A, Real Analysis.Since there are no solutions available for this problem set, I would like to show my attempt at a...
View ArticleCan every $C^2$ function defined on a closed set in $\mathbb{R}^d$ be...
When reading Page 147 of the book "Continuous Martingales and Brownian Motion" by Daniel Revuz & Marc Yor, I am confused with the Remark $3^\circ$ of (3.3) Theorem (Ito's formula).In this remark,...
View ArticleLeast number of circles required to cover a continuous function on a closed...
Now asked on MO here.This question is a generalisation of a prior question. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with radius $r$ required to cover...
View ArticleMeasurable in one variable and regular in the other
Suppose $(X,\mathcal{B},\mu)$ is a measure space and for $y\in\mathbb{R}$ suppose that the function $f(\cdot,y):X\to X$ is measurable and $f(x,\cdot)$ is continuous. In many cases, I have seen that one...
View ArticleSolving the transcendental equation $\operatorname{Li}_{3}(e^{-kx}) + x...
I need to solve the following equation:$$\operatorname{Li}_{3}(e^{-kx}) + x \operatorname{Li}_{2}(e^{-kx}) = k x^3$$for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here...
View ArticleHow to prove $\displaystyle\lim_{n\to\infty}{\frac{1}{\log n}}=0$?
For $n\rightarrow\infty$, $\log n\rightarrow\infty$. So, $\displaystyle\lim_{n\to\infty}{\frac{1}{\log n}}=0.$But I am unable to write the proof of it by using $(\epsilon, \delta)$ method.Please help...
View Article$f = g$ in $H^{-1}$ implies $f = g$ a.e.?
I have two functions $f, g \in H^1_0$. But I could only show $|f - g|_{H^{-1}} = 0$. Does that imply $f = g$ almost everywhere? I think this should follow immediately since if $f = g$ in $H^{-1}$,...
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