Counterexample of function that is hard to find
Is there a continuous function $f:\left [0,\dfrac{1}{2}\right ]\to \mathbb{R}$ for which there is no constant$c>0$ such that:$$|f(x)-f(y)|\leq\dfrac{c}{-\ln(|x-y|)},\ \forall\ x,y\in\left...
View ArticleTrying to prove monotone-sequences property ⇒ Archimedean property
Monotone-sequences property ⇒ Archimedean propertyToday I just started learning this and having trouble understanding parts of the proof. Sorry if these are easy to some, I just couldn't find this...
View Article$\phi$ is continuous and odd, $f$ is Lebesgue integrable, we define $T_nf =...
$\phi : \mathbb{R} \to \mathbb{R}$ is a continuous, odd function satisfying the conditions:$$\phi(0) = \phi(1) = 0, \ \ \ \ \ \ \phi(x + 2) = \phi(x) \ \forall x \in \mathbb{R}$$For a function $f$ that...
View Article$\sin(x) \leq \sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)!}$ when $n$ is even...
Let $n\geq 2$ be an even integer and $x\geq 0$. I want to show that$$\sin(x) \leq \sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)!}.$$Assume first that $x\in (0,\pi]$.By Taylor's theorem with Lagrange form...
View ArticleProof of associativity of addition for convergent infinite series in exercise...
As part of self-study, I am working on exercise 2.5.3 (a) of Understanding Analysis by Stephen Abbott (2015).The proof looks to me to be a simple application of the fact that subsequences of a...
View ArticleDoes there exist a sequence such that $\lim_{n\to\infty} \{(-1)^na_n^2\}=1$,...
Let $\{x\}$ denote the fractional part of $x$. I need an example of a sequence with all positive terms $(a_n^2)_{n\geq 1}$ such that $$\lim_{n\to\infty} \{(-1)^na_n^2\}=1\ \ \ \text{and}\ \ \...
View ArticleContinuity of the proper of a parameter-dependent integral
How to show that the integral of the discontinuous function $\operatorname{f}\left(x,y\right) = \operatorname{sgn}\left(x - y\right)$$$\operatorname{F}\left(y\right) =...
View ArticleDiscontinuous linear map and AC
The question arises when I am constructing an elementary proof for the following claim:Given a normed vector space $V$, the following are equivalent:$V$ is finite dimensionalEvery linear map $T:V\to V$...
View ArticleHow to prove that $A\subset B\implies \bar{A}\subset \bar{B}$
How to prove that A is a subset of B implies closure of A is a subset of closure of B?I encountered this problem when looking up the solution of Baby Rudin Chapter 2 Exercise 7:Let $A_1,A_2,A_3,\cdots$...
View ArticleTaylor expansion of...
I would like to compute the expansion of the following integrals near $N = + \infty$ up to $\mathcal{O}(1/N^2)$: $$ \int_{0}^{\omega_0} \frac{\sin \left( \frac{2 N + 1}{2} \omega \right) \cos(\omega...
View ArticleApplication of Banach contraction theorem to homotopy theory
Let $(X,d)$ be a complete metric space and $A$ be its closed subset.Suppose there is a homotopy $h: A \times I \rightarrow X$ between maps $f,g:A \rightarrow A$ such that$$h(x,0)=f(x), h(x,1)=g(x)$$...
View ArticleDefinite integral of Modified Bessel function, exponential and trigonometric...
I am trying to solve the following integral;$$ \int_{0}^{\frac{\pi}{2}} e^{\gamma \cos\theta} I_{1}(\epsilon\sin\theta)d\theta,$$where $\gamma\in\mathbb{R},\epsilon\in\mathbb{R}^{+},$ and $I_{1}$ is...
View ArticleWhy does $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)} = L \implies f =...
I am currently seeing a contradiction from my use of the "theorem"For any $2$ functions $f : \mathbb{Z}^{+} \rightarrow \mathbb{R}^{+}$ and $g: \mathbb{Z}^{+} \rightarrow \mathbb{R}^{+}$, if $\lim _{x...
View ArticleWhat is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} +...
We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$is $n$ (thanks to this question), butwhat is the global maxima of$\frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)},...
View ArticleAntiderivative of $\sin(\frac{1}{x})$, and computing the derivative of...
I am trying to understand Daniel Fischer's answer to this question Antiderivative of $\sin(\frac{1}{x})$I understand the ideas and every step except for the part where he saidThe first part...
View ArticleSame Subspace Closure implies Same Closure in Ambient Space?
$\DeclareMathOperator{Cl}{cl}$Let $\Cl_X$ be the map sending each set in a topological space $X$ to its closure $\overline{X}$. It is easy to derive the following identity:Let $S$ be a subspace of $T$,...
View ArticleBorel $\sigma$-algebra of $[a,b]$ and $\mathbb R$.
Let $\mathfrak B_{[a,b]}$ be the Borel $\sigma$-algebra of some non degenerate interval $[a,b]$ in $\mathbb R$. Let $\mathfrak B_{\mathbb R}$ be the Borel $\sigma$-algebra of $\mathbb R$. We want to...
View ArticleWhy the map $e^{A+B} = e^{A}e^B$ if $A,B$ are matrices that commute
Let $A,B$ be matrices with dimension $N$. Define $ e^A:= I+A+\frac{A^2}{2!}+.... = \lim_{n \to\infty} \sum_{k=0}^n\frac{A^k}{k!}.$Prove using limits if $AB = BA$ then $ e^{A+B} = e^Ae^B.$ I have...
View ArticleEvery non-empty perfect set in $\mathbb{R}^{k}$ is uncountable
In the Principles of Mathematical Analysis, 3rd Ed. by Rudin, he mentions in a Theorem 2.43 on page 41 that every non-empty perfect set $P$ in $\mathbb{R}^{k}$ is uncountable.In the proof, he assumes...
View ArticleProve that If $A \subset R^n$ a function $f : A \to R^m$ is continuous iff...
($\implies$) direction was easy enough, but I am confused on $(⟸)$. So we have that If $A \subset R^n$, for every open set $U \in R^m$ there is some open set $V \in R^m$ such that $f^{-1}(U) = V \cap...
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