Explain step in proof of $\lim_{n \to \infty} a_n = b \iff \limsup_{n \to...
I'm trying to understand a step in the forward direction of the proof of the theorem $\lim_{n \to \infty} a_n = b \iff \limsup_{n \to \infty} a_n = \liminf_{n \to \infty} a_n = b$.First, to clarify,...
View ArticleA countable intersections of derived sets
This question is from Hrbacek and Jech's intro set theory book (Ch 10, exercise 4.6). I am given:$$F=\{1\}\cup\{1-\frac{1}{2^{n_1}}-\frac{1}{2^{n_1+n_2}}-...-\frac{1}{2^{n_1+...+n_k}} : k\leq n_1...
View ArticleRegions whose area cannot be measure in the sense of Riemann
Consider a region $D$ of the plane. Usually, its area is defined as$$\iint_D 1$$Would it be possible that this integral does not exists, and thus $D$ has a non-measurable area?
View ArticleBernoulli inequality application
On some level of math in school we learn about Bernoulli's inequality. Proof of its correctness is very common in textbooks as exercise, when we learn mathematical induction.Is Bernoulli's inequality...
View ArticleShowing that convergence in measure does not imply convergence at at least...
When trying to show this I thought of the typewriter sequence but I couldn't find a reference for a full proof, just partial statements. Here is my attempt.We are working with the lebesgue measure on...
View ArticleDoes the Intermediate Value Theorem for uniformly continuous functions imply...
Proofs that the Intermediate Value Theorem (IVF) implies the Least Upper Bound Property for an ordered field usually use a continuous function that is not uniformly continuous like here...
View ArticleClassifying a second order non-linear ODE
I am currently dealing with the following ODE as a stationary, special case version of a PDE model derived from Kuramoto-Sivashinsky.$$y'' y' = ay$$Where $a$ is a real (constant) parameter.I am going...
View ArticleEvaluate the integral $\int_{-\infty}^{\infty}\binom{n}{x}dx$. [closed]
Evaluate the integral $\int_{-\infty}^{\infty}\binom{n}{x}dx$.This question came in Cambridge Integration Bee and I have no clue what to do in this.I rewrote $\binom{n}{x}$ as $\frac{n!}{{x!}{(n-x)!}}$...
View ArticleHow can we show that this integral is nonnegative?
Let$c_0>0$ and $\ell\in[0,1]$;$(E,\mathcal E,\lambda)$ be a measure space;$\mu$ be a probability measure on $(E,\mathcal E)$;$p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$p_\lambda:=\int...
View ArticleHardy littlewood maximal function of a function supported in the unit ball
This question comes from Stein’s Note on the class$L\log L$. In the proof for Theorem 1, which establishes for an integrable function $f\in L^1(\mathbb R^n)$ supported on a finite ball $B$, that the...
View ArticleUniform convergence on each compact subset implies uniform convergence in the...
Given an open region $D$, a sequence of functions $\{f_n\}$ analytic on $D$, and a function $f$ such that $\{f_n\}$ converges to $f$ uniformly on every compact subset of $D$, then the function $f$ is...
View Article$\psi,f:\Bbb R\to\Bbb R$ are continuous, $\psi=0$ outside $[0,1]$,...
The QuestionIf $\psi:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function $\ni$$\int_{0}^{1}\psi(y)dy = 1$, $\psi(y) = 0$ when $y$ doesn't belong to $[0,1]$ and $f: \mathbb{R} \rightarrow...
View ArticlePointwise convergence to absolute value function
I am trying to prove that,$$f_n(x) = x^{\large {1+\frac{1}{2n-1}}} $$converges point-wise to $f(x)=|x|$ for $x\in [-1,1]$My thinking was to prove that it converges point-wise by taking the limit of...
View ArticleFolland Real Analysis Problem 1.15
Problem Prove that if $\mu$ is a semifinite measure and $\mu(E) = \infty$, then for every $C > 0$ there exists $F \subset E$ with $C < \mu(F) < \infty$.My answer We can define a disjoint...
View ArticleThe oscillatory integral $\sup_{b, z > 0} \left| \int_0^b \frac{\cos(z...
I am trying to prove the boundedness of certain oscillatory integrals and I can not deal with the following situation, which I have reduced to a specific example. I claim that$$\sup_{b, z > 0}...
View ArticleIterative application of continuous function $f: \mathbb{R} \rightarrow...
The questionLet $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim_{n \to \infty}f^{n}(x)$ exists $\forall x$. Define $S = \{\lim_{n \to \infty}f^{n}(x): x \in...
View ArticleSandwiching the Lp norm sequence of random variable
Let X be a random variable with $\Vert X\Vert_p = E[\vert X\vert^p]^{1/p}<\infty$ for all $p\geq 1$.Assume for all $1\leq q < r < s$,$$\lim_{p\to\infty} \frac{\Vert...
View ArticleFind the derivative of $f(x)=\max\left \{ \min\left \{ \cos(x),\frac{1}{2}...
From my uni's textbook, classified as 'difficult'. Find the derivative (or right/left side derivative) of function $f$ on it's domain.$f(x)=\max\left \{ \min\left \{ \cos(x),\frac{1}{2} \right...
View ArticleLet $X \subset \mathbb{R}^{n}$ such that, for every compact $K \subset...
Let $X \subset \mathbb{R}^{n}$ such that, for every compact $K \subset \mathbb{R}^{n}$ , the intersection $X \cap K$ is compact. Prove that $X$ is closed.
View ArticleSimple counterexample for if $\lim_{x\to\infty}f(x)=\infty$ then $f$ is...
If $$\lim_{x\to\infty}f(x)=\infty,$$ then we can't conclude that there exists an $M$ where $f$ is monotonic on $[M,\infty)$ because $f(x)=\ln x+\sin x$ disproves that claim. What I want to know is, if...
View Article