Question on Kronecker lemma
Let $\{x_n\}_{n \geq 1}$ be a sequence in $\mathbb{R}$ and $\{a_n\}_{n \geq 1}$ be a sequence of positive real numbers satisfying $a_n \to \infty$ as $n \to \infty$. Further, suppose...
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Upper bound on the integral of a step function inequality
ProblemI have been working through content in real analysis and came across across an inequality in a proof that is unclear to me.We are told that a function $\psi$ is a step function on the interval...
View ArticleWhich is bigger, ${\sqrt2}^9$ or $9^{\sqrt2}$? (no calculator)
I was thinking about the classic question, "Which is bigger, $e^\pi$ or $\pi^e$?" (no calculator), and I tried to create a question that is similar, but resistant to the usual methods used for the...
View ArticleElliptic $L^1$-estimate
I know a version of elliptic $L^1$-estimate: If $\Omega \subseteq \mathbb{R}^n$ is a bounded open set and $f \in C_{c}^{\infty}(\Omega),$ then we have $$ \|f\|_{L^p(\Omega)} \le C\|\Delta...
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Let $ f(x)=\frac{1+\cos(2\pi x)}2$ and $f^n=f\circ f^{n-1}$. Is it true that...
Let $\displaystyle f(x)=\frac{1+\cos(2\pi x)}2$ for $x\in\mathbb R$, and $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$. Is it true that for Lebesgue almost every $x$, $\displaystyle\lim_{n \to \infty}...
View ArticleIf $A \subseteq \bigcup_{n\geq 1} A_n$, then $\lim_{n \rightarrow...
I'm trying to solve a measure theory problem. Here $\mu $ is a finite pre-measure in an algebra.To continue with the resolution, I opened the hypothesis that the following is true:If $A \subseteq...
View ArticleIf $f(x) + f(f(x))$ is linear, then is $f$ linear?
The question came to me after watching a video on YouTube: https://www.youtube.com/watch?v=_kVrljSYt_8I know that the function $g(x) = f(x) + f(f(x))$ is linear and therefore it holds that $g(ax+bx) =...
View ArticleProve the Existence of the Limit at Infinity for a Bounded Continuous...
I am working on a problem involving a bounded continuous function that satisfies a particular integral equation, and I aim to prove that the function has a limit as $ t \to + \infty $. Here are the...
View ArticleProving a variational inequality almost everywhere
Let $(\Omega_1,\mathcal{F}_1,\mathbf{P}_1)$ and $(\Omega_2,\mathcal{F}_2,\mathbf{P}_2)$ be two probability spaces (finite measure spaces). Suppose we have a random variable (measurable function) $X:...
View ArticleA barrier differential inequality implies a distributional differential...
I encountered the following statement while reading an article about Ricci flow:Thus $\Delta d_{K, p}^2\leq4$ in the barrier sense, hence in the viscosity sense and in the distributional sense.I...
View ArticleSemicontinious functions and hypographs
I've got a question about semicontinous functions and their hypographs. On wikipedia (https://en.wikipedia.org/wiki/Hypograph_(mathematics)) it is claimed that a function is upper-semicontious iff its...
View Articlestuck on proof (help with intuition): set $A$ is open $\iff A$ complement is...
So to prove $"\implies "$ I go by contradiction:Let $A$ be open and $A^c$ (the complement of $A$) also be open.since $A^c$ is open it does not contain its limit points.let $y$ be a limit point of...
View ArticleShow that $x+x^2\sin\frac{1}{x}$ is not increasing near $x=0$. [duplicate]
This problem is from the question given in my classes: we are asked to show that $h(x)=\frac{x}{2}+x^2\sin(\frac{1}{x})$ is not increasing near $0$, which can be proved by the fact that $h$ is $C^1$ on...
View ArticleContraction-like mapping without fixed point?
If $(X,d)$ is a complete metric space and $\xi:\;X\to X$ satisfies:$$d(x,y)<n+1\Rightarrow d(\xi(x),\xi(y))<n$$$$d(x,y)<1/n\Rightarrow d(\xi(x),\xi(y))<1/(n+1)$$for all $n= 1,2,\dots$, does...
View ArticleOrthonormal basis for a $L^2(I_1 \cup I_2)$.
We know that $\{e^{2 \pi i n t}\}_{n \in \mathbb{Z}}$ is an orthonormal basis for $L^2[0, 1]$. Suppose I have a union of two disjoint intervals $I_1 \cup I_2$. Consider $L^2(I_1 \cup I_2)$, does this...
View ArticleWays to define real multiplication and addition
The discussion below is mostly targeted towards teaching math to first semesters of an abstract mathematical major.I feel like teaching addition and multiplication in the usual way feels unsatisfactory...
View ArticleFind the convergence of the series $\sum \frac{n^{n-2}}{e^n n!}$
Show that the series $\sum_{n=1}^{\infty} \frac{n^{n-2}}{e^n n!}$ is convergent.I tried to use root test but it yields 1 which makes the test indecisive . Any other approach ?
View ArticleShow that $\sup \bigcup_{n=1}^\infty A_n= \sup_{n \geq 1}\sup A_n$
Let $A_n \subseteq \mathbb{R}, \forall n \geq 1$. If $A= \bigcup_{n=1}^m A_n$, then we know that $\sup A = \max_{1 \leq n \leq m}\sup A_n$.How about the infinite case, would be true that:$\sup A =...
View ArticleStrategies to find suitable function to apply MVT
1.Let $a, b$ be two positive numbers, and let $f:[a, b] \rightarrow \mathbf{R}$ be a continuous function, differentiable on $(a, b)$. Prove that there exists $c \in(a, b)$ such that$$\frac{1}{a-b}(a...
View ArticleIs the sequence $\sqrt[n]{\sum_{i=1}^n\cos^ni}$ convergent?
Does this sequence of number converge? If yes, what is its limit?$$a_n=\sqrt[n]{\sum_{i=1}^n\cos^ni}$$We know that $\displaystyle\sqrt[n]{\sum_{i=1}^n\cos^2i}$ is much simpler. This is an advanced...
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