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$A \subset [0,1]$. $P_a$ is parabola tangent to OX in $(a,0)$. $B = \bigcup_a...
$A \subset [0,1]$.$\forall_{a \in A}$ we name as $P_a$ a parabola that is tangent to $OX$ in point $(a,0)$.$B = \bigcup_a P_a \cap [0,a] \times \mathbb{R}$Show that:$$\lambda_2(B) = 0 \iff \lambda_1(A)...
View ArticleLeast number of circles required to cover a continuous function on a closed...
This question is a generalisation of a prior question. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with radius $r$ required to cover the graph of $f$?It is...
View ArticleIf $f(1)=f(2)=0$ then exist $c$ such that $cf''(c)+2f'(c)=0$
Let the function $f$ be continuous on $[0,2]$, second-order differentiable on the interval $(0,2)$ and satisfy the condition $f(1)=f(2)=0$. Prove that there exists a real number $c$ in $(0,2)$ such...
View ArticleProve $\lim_{n\to \infty}n/\log(n!)=0$ without using Stirling's approximation
It is convincing that the limit is 0 since $n$ has one fixed increasing step 1 but $\log(n!)$ has one strictly increasing step $\log n$. This is by seeing the limit as $\lim_{n\to...
View ArticleFind all polynomials in $\mathbb{P_n}$ such that the Fixed point iteration...
Determine all polynomial fixed point functions, $ p \in \mathbb{P_n}$ for some $n \geq 0$, thatsatisfy: Fixed point iterations on $p$ will always converge to some fixed point for any initial...
View ArticleMetric space with every infinite set having a limit point
I am trying to prove that a metric space in which every infinite subset has a limit point is compact. I am trying to prove it with Heine-Borel theorem for general metric spaces, since I have not...
View ArticleMultivariable limit...
I came today across the following problem:$$\lim_{(x,y)\to(0,0)}\frac{\ln(\cos(x^2+y^2))}{\ln(\cos(3x^2+3y^2))}$$My first thought was to substitute $$x^2 + y^2 = t$$ and solve the limit using...
View ArticleThe Medians of Lipschitz Functions on $(X,d,\mu)$ (Existence and Uniqueness)
Let $\varphi:(X,d,\mu)\to \Bbb R$ be a Lipschitz function, where $\mu$ is a probability measure on the metric space $(X,d)$. The median $m_\varphi$ of $\varphi$ is defined as the real number such...
View ArticleConvergence of $\sum\limits_{n=1}^\infty \sin \left( \frac{\sin...
I'm trying to figure out for which real values of $\alpha$ does this sum converges:\begin{equation}\sum_{n=1}^\infty \sin \left( \frac{\sin\left(\,{\sqrt{n}}\,\right)}{\sqrt{n^\alpha}}...
View ArticleRiemann-Integration: proof of sufficient condition for two variables functions
Let Ω⊂$\mathbb{R}^2$ be a regular domain, and $f$ : Ω→$\mathbb{R}$ a continuous and bounded function, with the exception of a set of discontinuous points with zero Peano-Jordan measure. Then $f$ is...
View ArticleCalculate this limit using integral
Giving $a$, $b$ are two positive numbers and $\alpha$ is a positive integer; calculate the limit:$$\lim_{n\to\infty}\left[\dfrac{1}{n^{2\alpha+1}}\sum_{k=1}^{n}\left(k^2+ak+b\right)^{\alpha}\right]$$I...
View Articlewhy is $L=\{\{x\mid x
Let the set $L$ be definded as$$L=\{\{x\mid x<q(i)\}\mid i\in\mathbb{N}\},$$where $q(i)$ is some bijection from $\mathbb{N}$ to $\mathbb{Q}$.Clearly, every member of $L$ is neither an empty set nor...
View Articlewhat's the nicest way to see that the sine power series is the trig definition
If one defines the sine function by its Taylor series, what's the easiest way to prove that this definition gives the trigonometry definition of opposite over hypoteneus?
View ArticleSpivak Calculus 3rd. Edition Chapter 1 Problem 12 (i) and (ii) Proofs Critique
Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next...
View ArticleDefinition of $\limsup$ from the right or from the left
Let $X$ be a metric space and $E \subset X$. If $f: E \rightarrow \mathbb{R}$, we can define for $a \in E$, the superior limit\begin{align}\limsup_{x \to a} f(x) = \lim_{r \to 0} \left( \sup\{ f(x) : x...
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$A \subset [0,1]$. $P_a$ is a parabola tangent to OX in $(a,0)$. $B=...
$A \subset [0,1]$.$\forall_{a \in A}$ we name as $P_a$ a parabola that is tangent to $OX$ in point $(a,0)$.$B = \bigcup_a P_a \cap [0,a] \times \mathbb{R}$Show that:$$\lambda_2(B) = 0 \iff \lambda_1(A)...
View ArticleContinuous function supported in $[2-\delta,2]$, satisfying integral of...
Fix $\delta >0$.I want to show the existence of continuous function $f$, satisfying $supp(f) \subset [2-\delta, 2]$ and $$\frac1{2\pi}\int_{2-\delta}^{2} f(x)\sqrt{4-x^2}\,\mathrm dx=1\;.$$How do I...
View ArticleIf every sub-sequence has a convergent sub-sub-sequence then the sequence...
Let $(X,d)$ be a metric space. Assume that $(x_n)$ is a sequence in $X$. There is a well-known theorem about the convergence of $(x_n)$ that reads as follows.Theorem. Every sub-sequence of $(x_n)$ has...
View ArticleHow can I prove that if $f:[0,+\infty)\to[0,1]$ is continuous and increasing,...
The problem is the following:Let $f:[0,+\infty)\rightarrow\mathbb{R}$ such that it is continuous, strictly growing and its image is contained in $[0,1]$, is uniformly continuous.I've tried in several...
View Article$\limsup_{n \to \infty} \Big|\frac{a_{n+1}}{a_{n}}\Big| > 1$ does not imply...
Question$\limsup_{n \to \infty} \Big|\frac{a_{n+1}}{a_{n}}\Big| > 1$ does not imply the divergence of $\sum a_{n}$.ContextIn baby rudin, chapter three, theorem3.34 says:-a.) If $\limsup_{n \to...
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