On real functions with the property that for each $\varepsilon>0,$ every...
Let $f:\mathbb{R}\to\mathbb{R}$ have the following property:$$\forall\varepsilon>0,\text{ no matter how large },\forall a<b,\ \exists a<x_1<x_2<b\text{ such that...
View ArticleReal life examples of pseudo-metrics
Every mathematician knows that, given a set $X$, a metric is a function $d:X\times X \to \mathbb{R}$ that verifies(i) $d(x,y)=0 \iff x=y$(ii) $d(x,y) = d(y,x)$(iii) $d(x,y)\leq d(x,z) + d(z,y)$From...
View ArticleUniqueness follows from explicit solution of $\ a + x = b\,$ in a group
I just started real analysis. I don't have a background in proofs or logic, simply calculus. So I'm trying to learn more about proofs--so forgive the basic question, please.How do you go about proving...
View ArticleOn the "idea" of calculus of variations to an inequality in the $L^p$ norm
Let C be a constant, satisfying $$\forall p \in [1,\infty),\, \forall x,y \in \mathbb{R}^n,\text{s.t.}\Vert x\Vert_p=\Vert y\Vert_p ,\,\text{and}\, \forall \theta \in [0,1], \, \\\Vert x-y\Vert_p\leq...
View ArticleApproximating Lipschitz Functions by $C^1$ functions
According to Evans-Gariepy as a corollary of the Whitney's Extension Theorem we have the followingTheorem (Approximating Lipschitz Functions)Suppose $f: \mathbb R^n \to \mathbb R$ is Lipschitz...
View ArticleFind the derived set of a set S [closed]
Our set $S = \{(-1)^n \left(1+\frac 1n\right)\,\text {for}\,n\in \mathbb N \}$.Find the derived set of the above set.
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Probability of exiting an interval from the upper end
I am struggling with computing the functional of a diffusion process. Assume that $(X_t)_{t \in [0,\infty)}$ is a one-dimensional, time-homogeneous, diffusion process such that the infinitesimal...
View ArticleProve or disprove that the sequence $x_n$ satisfying $|x_{n+1} - x_n| =...
I'm given a sequence of real nos. satisfying the condition$$|x_{n+1} - x_n| = \frac{1}{\sqrt{n}},\,\,\forall n \in \mathbb N$$I'm trying to prove that a sequence satisfying this condition will never be...
View ArticleFor $\{A_n\}$ a nested collection of open intervals $(a_n,b_n)$, must...
Let $\{A_n\}$ be a nested collection of open intervals $(a_n,b_n)$. Must the intersection $\bigcap_{n=1}^\infty A_n$ be nonempty?Briefly explain why is nonempty, or give an example where it is empty.
View ArticleAre observed values for random realized variables random themselves?
Regarding question What is a realization of random variable? and similarly Probability: are realizations of random variables what is actually observed?:If I have a random variable $X$, such that $X$ is...
View ArticleHow to solve $(a+b+c-x)^2 x \ge 4abc$, with $x \lt a+b+c$ and $x \in...
I'm interested in solving a non-trivial cubic inequality coming from some physical arguments.So, the first part of my question is dedicated to the creation of the (necessarily physical) context, the...
View ArticleConfusion on finding the algebra generated by the family $\{...
We need to find the algebra generated by this family, which means we need to find the smallest algebra containing the family. My approach is as follows. For any algebra $\mathscr{A}$ containing $\{...
View ArticleIf $\Sigma a_n$ is divergent , $b_n$ is unbounded and increasing then $\Sigma...
If $\Sigma a_n$ is divergent , $b_n$ is unbounded and increasing then $\Sigma a_n b_n $ is divergent ?I think it is true .Given $\Sigma a_n$ is divergent , so the sequence of partial...
View ArticleAre all types of infinite expansions of functions on a compact interval into...
If I expand a function $f(x)$ into an infinite sum of polynomials, according to any given set of rules, i.e. Taylor series, Legendre polynomials, or other types of polynomial expansion, if the function...
View ArticleBound for the upper level set
If $\int_X exp({a|f_j(x)|})d\mu(x)\le C$ for all $j$ (for some $a>0$) with $f_1,\dots,f_n$ on the probability space $(X,\mu)$ and $L^2$ norm of $f_j$ is bounded by $\eta^j$ for all $j$, for some...
View ArticleAbout the logical flow of proof of archimedean property
Recently, I have learnt about the archimedean property of real number system. part of the proof I found on textbook is the following. Assume archimedean property is wrong, which there doesn't exist any...
View ArticleIs there a name for the class of functions with at least one bounded derivative?
I am wondering if there is a name for the class of continuous functions $f:\mathbb{R} \to \mathbb{R}$, such that there exists an $n \in \mathbb{N}$ such that $|f^{(n)}| < C$ for some $C\in...
View ArticleThe proof that the integral definition does not depend on the particular...
A function $f: E \rightarrow \mathbb{X}$ is called simple if it can be represented as$$f(\omega)=\sum_{i=1}^k I_{E_i}(\omega) g_i$$for some finite $k, E_i \in \mathscr{B}$ and $g_i \in \mathbb{X}$.Any...
View ArticleConvergent series times a convergent sequence is a convergent series
If $\sum a_n$ converges and if $\{b_n\}$ is monotonic and bounded, prove that $\sum a_nb_n$ converges.I have a proof that involves the using partial sums, but would like to know if there's any other...
View ArticleConvergence or divergence of series $\sum\limits_{n=2}^{\infty} (\ln{n})^x$
Recently I'm reading K. F. Riley's book "Mathematical Methods for Physics and Engineering", I can't solve exercise 4.11(6) at page 145, and the solution/answer puzzles me.Question:Find the real value...
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