Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

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Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
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2answers
75 views

Find all solutions of $a^b = b^a$

Find all solutions to $$a^b = b^a$$ where $a, b$ are natural numbers with $a<b$. So far I've been able to conclude that this is equivalent to $$\frac{log(a)}{a} = \frac{log(b)}{b}$$ but I'm not ...
2
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0answers
95 views

Find the function whose Taylor series is $\log(x)+\log(x+1)+\log(x+2)+\ldots$

How do I find a function $f$ whose Taylor series is $$\log(x)+\log(x+1)+\log(x+2)+\ldots$$ for some point $x=a$? It would seem that $$\left.\frac{\partial^n}{\partial x^n}f(x) \right|_{x=a} = ...
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1answer
31 views

A Fubini-Tonelli's Theorem Problem

Let $E \subseteq \mathbb{R^n}$ a measurable set, such as for almost every $x \in \mathbb{R^n}$ we have $|E \triangle (E+x)|=0$ (Where $\triangle$ means simetric difference between two sets and ...
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18 views

Symmetrization Methods

I was wondering if I could get a list of the symmetrization methods out there i.e. methods that rigidly transform a set A into it's equimeasure ball $A^{*}$. Here are some: a) Steiner Symmetrization ...
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1answer
46 views

Find all the subsets bounded and nonempty such that sup(A)≤inf(A) [on hold]

Find all the subsets bounded and nonempty A such that $\sup(A)\leq\inf(A)$
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1answer
20 views

When $f(t)=ab(1-e^{-t})-c(e-e^{-t})>0$

Consider the function $f(t)=ab(1-e^{-t})-c(e-e^{-t})$, for $t>0$, where $a,b,c>0$. Can we find a sufficient condition on $a,b,c$ such that $f(t)>0$ for all $t>0$ ?
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23 views

Small question about limit at $+\infty$ to $-\infty$

please the definiton of $\displaystyle\lim_{u\rightarrow+\infty}G(t,u)=-\infty$ is: $\forall M>0 ,\exists R>0 $ such that $|u|\geq R \Rightarrow G(t,u)\leq M$ or $G(t,u)\leq -M$ ? please ...
5
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3answers
65 views

Is there a function almost everywhere $0$ on $\mathbb{R}$ whose graph is dense in $\mathbb{R^2}$?

Is there a function almost everywhere $0$ on $\mathbb{R}$ whose graph is dense in $\mathbb{R^2}$? How to establish such strange funciton?
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0answers
32 views

Flaw in the proof that a set is countable [duplicate]

Q: Let S be the set containing all sequences of 0's and 1's. i.e $S = \{(a_1,a_2,a_3,a_4,\ldots) : a_i = 0 \text{ or } 1\}$ Show that S is countable. Proof(Flawed) : Let $A_i$ be the ...
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34 views

$\{f_n\}$ is a sequence of continuous funcitons, then $\{x\in [0,1]: \sup_n |f_n(x)|=+\infty\}$ can't be like $[0,1]\cap \mathbb{Q}$

Let $\{f_n\}$ be a sequence of real valued continuous funcitons on $[0,1]$, then the set $E=\{x\in [0,1]: \sup_n |f_n(x)|=+\infty\}$ can't be like $[0,1]\cap \mathbb{Q}$. I think this question is ...
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0answers
23 views

Find limit property

Let $f$ be a function on $\mathbb{R}$ satisfy: $|f(x)-f(y)|\leq|x-y|$ $\forall x,y\in\mathbb{R}$. Consider the sequence: $$u_{n+1}=\frac{u_n+f(u_n)}{2},u_0=a$$ Research the limit property of this ...
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2answers
138 views

Is my proof that $\lim\limits_{n\to +\infty}\dfrac{u_{n+1}}{u_n}=1$ correct?

I'm doing an exercise where $(u_n)$ is a numerical sequence which is decreasing and strictily positive.While $(u_n)$ is a numerical sequence which is decreasing and strictily positive, then $(u_n)$ is ...
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1answer
36 views

Approximation functions in $L^{1}$ by indicator functions of dyadic cubes

Let $\mu$ be a finite positive regular Borel measure on $\mathbb{R}^{d}$ and let $S$ be the family of finite unions of squares of the form $\{a_{1}2^{n} \leq x_{1} \leq (a_{1} + 1)2^{n}, \ldots, ...
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0answers
60 views

Problem in functional analysis.

I heard of this problem that caught my attention and I am curious now thus I would appreciate if I could have a hint or a solution. Let $(x_n)$ a sequence in a normed space $X$ such that ...
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3answers
197 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
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1answer
42 views

Is the closure of the unit ball in $C^2[0,1]$ compact in $C^1[0,1]$?

Is the closure of the unit ball in $C^2[0,1]$ compact in $C^1[0,1]$? I don't know how to represent the ball in $C^2[0,1]$ with norm $\|\cdot\|_\infty$ or some other norm, I think this problem should ...
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1answer
28 views

alternating series test convergence proof with Cauchy criterion

If we have a sequence $(a_n)_{n=1}^{\infty}$ and this sequence is decreasing and converge to 0, how can I show that the sequence $s_n = a_1 - a_2 + a_3 - a_4 + a_5 + (-1)^{n+1} a_n$ of alternating ...
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2answers
20 views

Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
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1answer
44 views

Limsup of intervals $(-\infty, - n \sin(1/n))$

I think that $\limsup \big (-\infty, - n \sin \frac{1}{n} \big )$ (as the lim sup of sets) is $(-\infty, 1)$. Is this correct? If so, how do we prove it? Background: I'm trying to do a question in ...
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104 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
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1answer
29 views

Convergence of a subsequence .

If every subsequence of $x_n$ has a further subsequence which converges , is it true that the sequence is convergent? NOTE : This is not a duplicate ofthis . In this problem it is not given that the ...
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0answers
106 views
+50

Question on differentiate under integral

First we have the following theorem: Then we apply it to a concrete problem: Finally how to obtain the second rectangle?
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56 views

Question on differentiation under the integral

How to obtain the red rectangle? Why factor n is disappeared?
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2answers
58 views

$\delta-\epsilon$ proof for the limit $\lim\limits_{x\to3^-}\sqrt{3-x}=0$. [closed]

I have no idea how to start in this question since it is a left hand limit. Thanks for any help! :)
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0answers
98 views

Definite trigonometric integral

This question is motivated by Iterative Mean, Covariance Algorithm Convergence: Is there a closed form for the integral $$ \int_0^{2 \pi} ...
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2answers
352 views
+150

Understanding this ODE

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
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1answer
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The Landau symbol $\mathcal{o}$ as in Königsberger Analysis I

I am currently working on Chapter 14 - local approximations of function and Taylor polynomials - in Königsberger Analysis 1 Background: Königsberger introduced the Taylor Polynomial of order ...
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1answer
33 views

Surface integral over d-sphere for $|x-y|^{(2-d)}$

I am looking for $\int_{S_{r}(0)}|x-y|^{2-d}dS_{y}$ for $x\neq 0$. The parametrization is hard to work with and the integrand is not rotationally symmetric. I will post any updates. any ideas thank ...
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2answers
50 views

Set of analyticity of a function is an open set

Let be $f$ a real function of real variable, prove that the set in which $f$ is analytic is an open set. Any help please?
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79 views

Derivative in 0

I'm a highschool student and we don't learn maths in English. So please excuse me for my Math's English. I'm doing an exercise and I can't answer its final question. Can you help me? Thank you! Let ...
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1answer
19 views

Lipschitz continuous set-valued map

Let $X$ and $Y$ be metric spaces and $F:X\to2^Y$ be a set-valued map. Suppose that $2^Y$ is endowed with the Hausdorff metric. I wonder about sufficient conditions on $F$ that ensure this map is ...
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1answer
47 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$ [closed]

Let $|x_n|<1$. Is the given series convergent or which conditions do we impose on the sequence $(x_n)$ to given series be convergent? $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$
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3answers
103 views

Find $\delta >0$ such that $\int_E |f| d\mu < \infty$ whenever $\mu(E)<\delta$

I am studying for a qualifying exam, and I am struggling with this problem since $f$ is not necessarily integrable. Let $(X,\Sigma, \mu)$ be a measure space and let $$\mathcal{L}(\mu) = \{ \text{ ...
3
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0answers
61 views

Upper bound for the sums of powers of factors

Fix $\alpha \in \,]0,1]$. Is it true that for each sufficiently large positive integer $n$, if $n = x_1 \cdots x_j$, for some integers $x_1, \ldots, x_j \geq 2$, with $j \geq 2$, then $$x_1^\alpha + ...
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1answer
19 views

Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$? [closed]

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$?
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4answers
124 views

Prove $f(x) < 0 \forall x$

Let $f$ is a function from $\mathbb{R}^+$ to $\mathbb{R}$ twice continuously differentiable and: $f''(x)\leq f(x)$ $\forall x$ $f(0)=0$ $f'(0)=0$ How can I prove $f(x)\leq 0 $ $\forall x ...
3
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1answer
35 views

Injective function, continuous at $x$, not locally monotone at $x$.

I set out to prove the following statement or give a counterexample: Suppose $f:[a,b] \to \mathbb{R}$ is one to one. Suppose $f$ is continuous at $x\in [a,b]$. Then there is a neighborhood of ...
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1answer
48 views

Is $(|x_n|^{n+1})$ convergent or bounded if $|x_n|<1$?

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(|x_n|^{n+1})$ convergent or bounded?
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41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
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231 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
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1answer
42 views

Proof of Differentiate under the integral

there are four conditions but I want to know where condition $(i)$ is uesd?
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1answer
71 views

Proof about Riemann integrability of a bounded function

I tried to prove the following, please could somebody tell me if my proof is correct? If $f: [a,b]\to \mathbb R$ is a bounded Riemann integrable function then for every $\varepsilon > 0$ there ...
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2answers
47 views

Is there a (linear) functional $f\in (C[-1,1])^*$ such that $f$ maps even continuous function to $0$ and odd continuous function to its infinite norm

Is there a (linear) functional $f\in (C[-1,1])^*$ such that for any even function $g$ in $c[-1,1]$, $f(g)=0$, and for any odd fucntion $h$ in $c[-1,1]$, $f(h)=||h||_\infty$?
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What is $\int f$ if $f$ is not Riemann integrable in the reverse direction of this theorem

Consider following theorem: I wanted to prove the $\Longleftarrow$ direction when I run into trouble. I do not understand the expression $|R(f,P)-A|$. Here $R(f,P)$ is a Riemann sum with respect to ...
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0answers
32 views

Singular measures

Suppose $\mu$ and $\nu$ are two signed measures and they are mutually singular. I heard 2 version on this definition there $\exists N $ s.t.$\mu(N)=0$ and (1) $\nu(N^c)=0$ (2)$|\nu|(N^c)=0$ which ...
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0answers
89 views

Does this arithmetic operation have a name

I've came across the following product-like operation on reals $$ a\times b:=1 - (1-a)(1-b). $$ This operation is commutative, associative and has $1$ as a zero element and $0$ as a unit element: $$ ...
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1answer
40 views

A problem in the proof of Jordan decomposition theorem

How to obtain the red rectangle in the picture? thanks in advance.
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1answer
36 views

Denseness and non-measureability

I made up a question for myself and tried to answer it. I'm not completely sure of my question and answer, since I lack a grounding in analysis (the tragedy of doing physics then mathematical ...
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1answer
46 views

A question of real analysis.

If for every $\epsilon>0$ there exists $\delta>0$ such that $$|f(x)-f(1)|\geq\epsilon $$ whenever $$|x-1|\geq\delta$$ Then which of the following is true? A.$f$ is discontinuous at $x=1$ B.$f$ ...