dist$(x,A)=0$ if and only if $x\in \overline{A}$ (closure of $A$)
Let $(X,d)$ be a metric space and non-empty $A\subset X$. The distance from $x$ to $A$ is defined as $$\mbox{dist}(x,A)=\inf\{d(x,a):a\in A\}$$I want to show that dist$(x,A)=0$ if and only if $x$ is in...
View ArticleA sum with strong oscillations
I encounter a sum in the form\begin{equation}\sum_{j=1}^{\infty}a_{j}b_{j}e^{i j\theta}\end{equation}I expect a lot of cancellations to happen due to the oscillatory term $e^{ij\theta}$.I can control...
View ArticleHelp me show that $\max_{k \leq n}|X_k|$ is a martingale
Related to this questionSetup$(\Omega, F, P)$; probability spaceLet $\{F_n\}_{n=0,1,2,\ldots}$ be a filtration, and $X=\{X_n\}_{n=0,1,2,\ldots}$ be an adapted process.In addition, we define$$X^\ast_n =...
View ArticleTesting for convergence of a series
Test the following series for convergence$$\sum_{k=1}^\infty \sin k \sin \frac{1}{k}$$.I am just wondering if the following method is ok:$$\sum_{k=1}^\infty \sin k \sin \frac{1}{k} = \sum_{k=1}^\infty...
View ArticleExistence of global minimum of convex function
Let $f:I\longrightarrow \mathbb{R}$ a convex function cotinous on a interval $I$.If $f$ not is monotone, then is true that $f$ has a global minimum in inner of $I$?--My attempt: because $f$ is no...
View ArticleIs this functional in equality $\left(\int_0^1 |f'(x)| dx\right)^2 \geq C...
Inspired by Sobolev inequalities, I would like to find a universal constant $C>0$ such that$$ \left(\int_0^1 |f'(x)| dx\right)^2 \geq C \left(\int_0^1 f(x)dx -1\right) $$for all functions $f \in...
View ArticleProve that every well defined function satisfies $\epsilon-\delta$ definition...
The motivation for this question is a comment on this post, we have:$$ ∀\epsilon ∀x ∃δ(x): |x−x_0|<δ(x)⟹|f(x)−f(x_o)|<ϵ$$With epsilon, delta being in $\mathbb{R_{\geq 0}}$ and $f: D \to...
View ArticleRudin's proof on the Analytic Incompleteness of Rationals [duplicate]
In Rudin's classical "Principles of Mathematical Analysis," he gave a proof like this:Claim: Let $A= \{p\in \mathbb{Q} | p>0, p^2 <2\}$. Then A contains no largest number.Proof: Given any $p\in...
View ArticleOn predictability of deterministic processes and approximation of Borel...
Let $(\Omega, (\mathcal F_t)_{t\geq 0}, \mathcal F)$ be a filtered measurable space. Consider the deterministic process $X(t,\omega) = h(t)$ where $h: \mathbb R_+ \rightarrow \mathbb R$ is a function....
View ArticleReferences for multivariable calculus
Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with...
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Finding the conditions that $a,b,c$ should satisfy s.t....
Recently, I encountered a problem related to finding the roots of an equation, which has troubled me for quite some time. The problem can be described as follows:Given that $a,b,c$ are all arbitrary...
View Article$-p\log p \prec H(p)p$: $H(p )p$ majorizes $-p \log p$ where $H(p)$ denotes...
$$ \color{blue}{-p\log p \prec H(p)p}.$$I guess that above majorization relation holds for any probability vector $p=(p_1,\dots,p_n)\ge 0$ with $\sum_{i=1}^np_i=1$, where $p\log p=(p_1\log...
View ArticleProve that if the numbers $a$, $b$, $c$, $d$, $n$ are natural numbers, this...
Prove that if the numbers $a$, $b$, $c$, $d$, $n$ are natural numbers, this equality is impossible: $$a^2+b^2+c^2+d^2-4\sqrt{abcd}=7\cdot2^{2n-1}$$.The only thing I did was convert the left side to...
View ArticleProve $\mathrm{argmax}_{i} \frac{x_i}{L+y_i}$ is same as of...
Suppose $x$ is a set such that $x = [x_1, x_2, \dots]$ and $x_i \geq 0$, also let $y = [y_1, y_2, \dots]$ and $y_i>0$. How can I prove that$$\mathrm{argmax}_{i \in \{1,2,\dots\}} \frac{x_i}{L+y_i} =...
View ArticleFinding all $x+y+z$ for positive integers satisfying...
Let $x, y, z$ be positive integers such that$$\frac{13}{x^2}+\frac{28}{y^2}=\frac{z}{85}$$Find all $x+y+z$.I observe that $(x, y)=(1, 1)$ or $(1, 2)$ will give some solutions. However I can neither...
View ArticlePartial derivatives of polynomial in two variables
Let $k \in \mathbb N$, $a_{ij} \in \mathbb R$ for $i,j \in \mathbb N$, $i+j \le k$. A function $f:\mathbb R^2 \to \mathbb R$$$f(x,y) = \sum_{i+j \le k} a_{ij}x^iy^j$$is called polynomial of degree $k$...
View ArticlePossible limits of subsequences in the ratio test
Let $0<y_{n}$ be a sequence such that $\frac{y_{n+1}}{y_{n}}\to C$for some $0<C<\infty.$ Let $x_{k}=y_{n_{k}}$ be a subsequence of$y_{n}$ such that $\frac{x_{k+1}}{x_{k}}\to B,$ for some...
View ArticleShow that $\sum_{n \in \mathbb{N}} \frac{(-x)^n}{n(1 + x^n)}$ converges...
I have tried to use abel's test and solve as followingTake sequence ${b_n(x)}$ as $\frac{-1^n}{n}$ which is uniformally convergent by Lebinitz test. Also $\frac{x^n}{x^n+1}$ = $\frac{1+x^n-1}{1+x^n}$ =...
View ArticleIf $(f_n)$ is uniformly limited, $f_n\rightarrow_u f$, $(g_n)$ is...
Let $f_n:\mathbb{R}^m\rightarrow\mathbb{R}$ and $g_n:\mathbb{R}\rightarrow\mathbb{R}$ be sequences. Suppose that:$(i)$$(f_n)_n$ is uniformly limited and $f_n\rightarrow f$ uniformly, for a certain...
View ArticleProduct of a concave function and a decreasing function
Let $f(x)$ and $g(x)$ be two positive continuous functions.Function $f(x)$ is concave at $x_1$ and function $g(x)$ is $<1$ and is decreasing; $g(0)=1$ and $f(0)=0$.Define function $h$ as the product...
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