2 Commuting, Continuous mappings from a closed interval onto itself doesn't...
The question is as such:If two continuous mappings $f$ and $g$ of a closed interval into itself commute, that is, $f\circ g=g\circ f$, then they do not always have a common fixed point.-- Zorich...
View ArticleProve that the sequence is monotonically decreasing
Give the sequence:$a_1 = 1, a_{n+1}=72/(1+a_n)$In Thomas' Calculus: Early Transcendentals, exercise 10.1 question 91, they asked to assume that the above sequence converges and to evaluate the...
View ArticleProve that $F[f](y) = o(1/y)$
I'm reading by calculus lectures and there is the following task:How can we prove that if $f$ is even, strictly monotonically decreasing on $[0, \infty)$ and $f \in C^1(\mathbb{R})$ then it's Fourier...
View ArticleProof Verification: Supremum and Infimum of $A=\{n+\frac{(-1)^n}n|n\in\mathbb...
ProblemIf $A=\{ n + \frac{(-1)^n}{n} | n \in \mathbb{N}\}$, prove that $A$ is not bounded above but bounded below and $\inf(A)=0$.My answerLet $A=B\cup C$ where $B=\{ n - \frac{1}{n} | n \in...
View ArticleIs every continuum-sized dense subset of the irrationals order isomorphic to...
This is a strengthening of a question another user asked, here: Are irrational numbers order-isomorphic to real transcendental numbers?. In the answer to that question, it was stated that the...
View ArticleSolution verification of: $f$ periodic and $\lim_{x \to \infty} f(x)$ exists...
Let $f$ be a periodic function. Show that if $\lim_{x \to \infty} f(x)$ exists in $\mathbb{R}$, then $f$ is a constant function. Use this result to prove that $\lim_{x \to \infty} \sin x$ doesn't...
View ArticleClik here to view.
Prove that exponentiation is continuous
This is from Tao's Analysis textbook. Let $𝑎>0$ be a positive real number. Then the function $f: \Bbb{R} \to \Bbb{R}$ defined $𝑓(𝑥):=𝑎^𝑥$ is continuous. It is hinted that we utilize the given...
View ArticleDyadic Expansion-Proof?
Working through a measure theory textbook, and would like to understand dyadic expansions before I can understand its connections with the law of large numbers. I want to see this proved in detail,For...
View ArticleA question on Group of homeomorphism of $[0,1]$.
Consider $[0,1]$ with the usual euclidean topology. Now $G$ be the set of homeomorphisms from $[0,1]$ onto $[0,1]$. $G$ forms a group under composition. Now, Let $F=\{ f\in G | f(0)=0 \}\ $. Now $F$ is...
View ArticleCalculate the integral of $\frac{xe^{-3x}}{(3x-1)^2}$. [closed]
Calculate the integral of $\frac{xe^{-3x}}{(3x-1)^2}$. I introduced a new variable $t = 3x - 1$ and obtained $\frac{1}{9} \int \frac{u+1}{u^2} e^{-(u+1)} du$. Now I have split the integral into two...
View ArticleConvergence/Divergence of Recursive Sequence
Let $a_1=1$ and $a_{n+1}=\frac{a_n(\sqrt n+\sin n)}n$ and $b_n=a_n^2$. Need to show convergence of $a_n$ and $b_n$.I tried solving it as below, but I am not sure about it.My approach as follows:$ -1/n...
View Articlereal quasi-convex functions
A function $f:I\longrightarrow \mathbb{R}$ with $I$ inteval is said quasi-convex if $\forall \alpha\in\mathbb{R}$ the set $\{x\in I:f(x)\le \alpha\}$ is a interval.Equivalently $f(tx+(1-t)y)\le...
View ArticleMajorization and second largest probability
I have a vector $p_1\geq p_2\geq\cdots p_m\geq0$ such that $\sum_{i=1}^mp_i=1$. Majorization is defined as follows: For vectors $ x, y \in \mathbb{R}^n $, $ x \prec y $ if $\sum_{i=1}^k x_{[i]} \leq...
View Article$\sinh(x^n)\le \sinh^n(x), \forall x\in[0,1], n\in\mathbb{N_{>0}}$
I want to prove that:$\sinh(x^n)\le \sinh^n(x), \forall x\in[0,1], n\in\mathbb{N_{>0}}$I tried proving this inequality using induction, but I can't prove that the statement is true for $n+1$. Is...
View ArticleUnderstanding of Rudin Definitions 9.6c: $|A \mathbf{x}| \le...
Let $A$ be a linear transformation of $\Bbb{R}^n$ into $\Bbb{R}^m$.Define the norm $\|A\|$ of$A$ to be the sup of all numbers $|A \mathbf{x}|$, where $\mathbf{x}$ranges over all vectors in $...
View ArticleProving a lower bound for $\iint_{\mathbb...
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as$$T_k(t)=\begin{cases}t-k &...
View Article$L^{\infty}$ (uniform) decay of Dirichlet heat equation $u_t=\Delta u$
It is known that the heat PDE$$\frac{\partial u}{\partial t}(t,x)=\Delta u(t,x)$$ on a smooth bounded open subset $\Omega\subset\mathbb{R}^N$ and $u=0$ on the boundary $\partial \Omega$, has a smooth...
View ArticleWhat are some useful problem solving strategies for real analysis?
In this blog, Professor Tao exhibited some problem solving strategies that can help students in their study of (mostly) measure theory and some are intended for analysis in general. I'd love to see...
View ArticleExercise 7.1.1 - Analysis I - Terence Tao
I have a problem, while proving part (b) of lemma 7.1.4.We wish to prove the following:(b) Let $m \leq n$ be integers, $k$ be another integer, and let $a_i$ be a real number assigned to each integer $m...
View ArticleAdjacent vs. All-Possible Distances in the Definition of a Cauchy Sequence
I'm currently learning about Cauchy sequences, and I'm trying to build my intuition regarding its definition. My question is on how we can motivate why we consider the distances between all points and...
View Article