Supremum of a nonempty set bounded from above is unique.
Proof. Let $A$ be nonempty and bounded from above. Let $s_{1}$, $s_{2}$ be two supremums of A. Since $s_{1}$ is an upper bound and $s_{2}$ is less than equal to any upper bound. We have $s_{2}\leq...
View ArticleFor which sequence does this summation method converges fast to $0$?
For a bounded sequence $(a_k)_{k=1}^\infty$ define the sequence $(A_j)_{j=1}^\infty$ by$$A_j = \sum_{k=1}^\infty \frac{a_k}{k^2+j^4}.$$It is elementary to check that (I cheated and used...
View Articleapplication of Implicit function theorem to show existence of a solution of a...
Can we deduce anything about existence of a solution of a differential equation using Implicit function theorem. I feel we can but I am unable to setup things to apply implicit function theorem.Let's...
View ArticleCan we get rid of monotone property?
Given $f$, $f_{1}$, $f_{2,...}$ are monotone functions on $(a, b) \subset \mathbb{R}$ and $f_{n}$ converges to $f$ in the Lebesgue measure on$(a, b)$. Prove that $f_{n}(x) \to f(x)$ at every point $x$...
View Articlelimit of infinite seqences
I am trying to prove this below proposition from real analysis royden pg 23 4th edition.Let $\{a_n\}$ and $\{b_n\}$ be real sequences, if $a_n <= b_n$ then $\limsup\{a_n\} <= \liminf\{b_n\}$but I...
View ArticleHow find this inequality minimum...
Let $n$ be a given positive integer, and let $a_{1},a_{2},\cdots,a_{n}\ge 0$ such that $a_{1}+a_{2}+\cdots+a_{n}=1$.Find this minimum...
View Articlewhat is the equation between A'x and A'y？
A is at (0,0),B is at(200,0).B is going to do a motion of free fall(g is 10),A's speed is 60 and always towards B.enter image description herethe question is1.can A hit B?2. What is the equation of A'x...
View ArticleInequality by induction $ n^2 \leq \left( \sum_{j=1}^n x_j \right) \left(...
I am trying to prove the following inequality: for all $n \geq 1$ and all $x_1, \ldots, x_n \in \mathbb{R}^{*}_+$, one has$$n^2 \leq \left( \sum_{j=1}^n x_j \right) \left( \sum_{j=1}^n \frac{1}{x_j}...
View ArticleCayley transform from CR sphere and Heisenberg group
The CR sphere $\mathbb{S}^{2 n+1}$ is the boundary of the unit ball $B$ of $\mathbb{C}^{n+1}$. In coordinates, $\zeta=\left(\zeta_1, \ldots, \zeta_{n+1}\right) \in \mathbb{S}^{2 n+1}$ if and only if...
View ArticleWhy this function well defined?
Let $M^{n-1} \subset \mathbb{R}^n$ be a smooth compact manifold, $\Lambda$ = $\{U_i\}_{i \in I}$ be an open cover of $\mathbb{R}^n$. Suppose that for every $U_i$ there is a smooth bounded function $f_i...
View ArticleProve that $[a,b]$ is compact
Let $a<b$ be real numbers. Prove that $[a,b]$ is compact.Below I present my solution. I thought it's good enough, but my TA said it's incorrect. I don't see where there is a problem. Could you help...
View ArticleShow that for non-negative measurable functions $f,g$ with $fg \geq 1$ the...
Let $\mu(\Omega)$ be a probability measure (i.e. $\mu(\Omega) = 1$), and let $f,g$ be non-negative measurable functions on $\Omega$ such that $fg \geq 1$. Show that $1 \leq (\int f^p )(\int g^p)$ for...
View ArticleRiemann-integrability of a piecewise function, similar to Thomae and...
Consider function $f $ defined on an interval $[-1,1] $. Evaluate whether $f $ is continuous, Riemann integrable and/or Newton integrable on the given interval.$f(x) = \left\{\begin{matrix}x ,&...
View Article$L^{p}_{loc}$ as a normed space
What norms can we define on $L^p_{\mathrm{loc}}$ ?or What is the most commonly used norm on $L^p_{\mathrm{loc}}$.It is tempting to define $$\|f\|_{L^p_{\mathrm{loc}}}:=\sup_{K\;\text{is...
View ArticleWhittaker function and gamma function
I am not an expert in special function. I would like to know if one of the following expressions can be simplify in some way, maybe using some other special functions (I tried with Coulomb wave...
View ArticleLimit of a sequence without using log
For $n=2,3,4,\dots$, define$$r_n=\min\left\{r\in\mathbb Z\colon\, r\geq 1\,\text{ and }\,\frac{1}{r}+\frac{1}{r+1}+\dots+\frac{1}{n-1}\leq 1\right\}.$$Consider the sequence$$s_n=\frac{r_n}{n}.$$One can...
View ArticleIdentify the closure, interior and boundary of the next convex set
I must find the boundary, closure and interior of this set\begin{equation*}S = \{\mathbb{x}: x_1 + x_2 \leq 5, -x_1 + x_2 + x_3\leq 7,x_1,x_2,x_3\geq0 \}\end{equation*}For the closure, i plan to proof...
View ArticleIf $f$ is a strictly increasing function with $\lim_{x\rightarrow...
Let $f:[0,\infty)\rightarrow \mathbb{R}_+$ be a strictly increasing continuous function with $f(0)=0$ and such that $\lim_{x\rightarrow \infty}f(x)=\infty$. I want to check if $\lim_{y\rightarrow...
View ArticleDeterminants based questions which contain singular and non singular matrices
if A and B are real matrices of order and such that det(A).det(B) < 0 then the set of matrices over x belong to R given by C(x) = Ax +B is singular for at least one x belong toR and at most n values...
View Article$L_{p}$ spaces with pointwise multiplication
Are any of the $L_{p}$ spaces Banach algebras if multiplication is defined pointwise; that is, $(fg)(x)=f(x)g(x)$?
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