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, and integration through the fundamental theorem of calculus.

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Deducing this inequality from Cauchy-Schwarz?

I need to prove that for all $n \in \mathbb{N}$ we have the inequality $$ \sqrt{\sum_{k=1}^n (x_k - y_k)^2} \leq \sqrt{\sum_{k=1}^n (x_k - z_k)^2} + \sqrt{\sum_{k=1}^n (z_k - y_k)^2}. $$ The hint ...
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1answer
13 views

Extension of Fourier transform to $L^2(\mathbb{R})$

We defined the fourier transform and it's inversion for the Schwartz class $S(\mathbb{R})$. Since $S(\mathbb{R})$ is dense in $L^2(\mathbb{R})$, we can find for a given $f\in L^2(\mathbb{R})$ a ...
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2answers
15 views

For any measurable set $A\subset\mathbb{R}$ and $r\in(0,\mu(A))$ we have $(\mu|_{2^A})^{-1}(r)\neq\emptyset$

Recently when I tried to prove a statement I needed to rely on the following fact that intuitively feels correct, but I wasn't able to prove it accurately. Here it is: Consider a set $A\subset\...
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0answers
8 views

Upper sum is strictly subadditive

Can you give me an example demonstrating that the upper sum is strictly subadditive (i.e. (the upper sum of $f$) + (the upper sum of $g$) is strictly bigger than the upper sum of $(f+g)$)?
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1answer
20 views

Intersection of a nested interval of $A_n=\left[3-{\frac{1}{\sqrt{n}},3+\frac{1}{3^n}}\right]$

$A_n=\left[3-{\frac{1}{\sqrt{n}},3+\frac{1}{3^n}}\right]$ What is $\bigcap_{n=1}^{\infty}A_n$ Since every set becomes a subset of the next set, is it correct to say that the intersection of all ...
4
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1answer
51 views

Proving that a function is integrable

Given that $f:[0, \infty] \to \mathbb{R}$ is decreasing with $\displaystyle\lim_{x \rightarrow \infty} f(x)=0$, prove that $$I=\int_{0}^{1}\frac{\cos(\frac{1}{x})f(\frac{1}{x})}{x^2}dx$$ converges. ...
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4answers
38 views

Find a counter example

The interior of the union is the union of the interiors. $\text{int}\left(A\cup B\right) = \text{int}(A) \cup \text{int}(B)$ I'm not too sure about to get started with this one. Any hints so as to ...
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1answer
16 views

Necessary and/or sufficient conditions for summability of a sequence

It is clearly true that any $(a_n)_{n=1}^\infty$ that has $$a_n=O(n^{-1-\varepsilon}),$$ for some fixed $\varepsilon>0$, is absolutely summable: $$\sum\limits_{n=1}^\infty |a_n|<\infty.$$ My ...
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5 views

$e^{-x^2}M_{k,m}(x^2) \in L_1$ space?

Let $M_{k,m}(z)$ be m-Whittaker Function defined in enter link description here. How can show that $e^{-x^2}M_{k,m}(x^2),\, m,k > 0$ is belongs to $L_1\left(\mathbb{R}\right)$ or not? Thank you ...
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24 views

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective?

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is a continuous function. I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(...
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0answers
11 views

Trobule with an application of Green's representation formula

The teacher solved an exercise in class which required you to prove that, if $\Omega$ is a bounded domain in $\mathbb R^n$ and $G$ its Green function, then $G$ is symmetric, i.e. $G(x,y)=G(y,x)$ for ...
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0answers
9 views

Problem in understanding the proof of 'Limit of Image of Sequence'. [duplicate]

So, I was following the proof of 'Limit of Image of Sequence'; here is the theorem: Let $M_1=(A_1,\mathrm d_1)$ and $M_2=(A_2,\mathrm d_2)$ be metric spaces. Let $f:A_1\mapsto A_2$ be a ...
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1answer
21 views

Showing a function is integrable

Let $\xi, \zeta\in\mathbb{R}^m$. How might one try to show that $\displaystyle\frac{1}{(1+\left|\xi - \zeta\right|)^{k}}$ is integrable for large $k$? I can see that the problem occurs on the "...
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2answers
43 views

How can we say the derivative is exact if the difference quotient has a domain restriction?

I think I've finally been able to voice my confusion when it comes to derivatives and limits. Let's first look at the difference quotient for a function $f(x)=x^2$ $$\lim_{h\to0} \frac{f(x+h)-f(x)}{...
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1answer
35 views

Are minimizing a function and root finding the same?

What is the relationship between minimizing a function and finding a root of an equation? Are the the same? I know in both problem we have similar algorithms, such as gradient decent, or newton's ...
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0answers
19 views

Creating a tight frame of $\mathbb{R}^{n}$ when already knowing some of its vectors.

I'm wondering whether or not it's possible to start with a matrix $S\in\mathbb{R}^{m\times mn}$, $m<n$, and add rows to it so that the columns of the resulting matrix form an orthogonal system of ...
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2answers
23 views

I have to maximize this function involving absolute values

f(x) = $\frac{1}{1+|x|}$ + $\frac{1}{1+|x-2|}$ needs to be maximized. Maximizing this function means minimizing the denominators simultaneously. So I have to find the minimum value of 1+ $|x|$ and 1+...
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0answers
16 views

shifting integration variable and taking derivative seemingly giving problem

I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example. I have an integral that looks approximately ...
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0answers
33 views

Problems with the Banach fixed point theorem

At the moment, I am studying for an exam and I came across the following exercise: Consider the map $f:[0,1] \to \mathbb{R}$, $f(x)=1-\arctan(x)$. Prove the following statements: a) $f$ has a ...
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0answers
31 views

How does one visualize Abel's test?

I remember that at a lecture we had a visual representation of why Abel's test should work but I don't recall how exactly it was. It was somehow representing $a_{n}$ on one axis and $b_{n}$ on another ...
2
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1answer
38 views

What is the difference between hyperreal numbers and dual numbers

Wikipedia has two different but unconnected pages for Hyperreal and Dual numbers. https://en.wikipedia.org/wiki/Hyperreal_number and https://en.wikipedia.org/wiki/Dual_number I cannot stop seeing ...
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1answer
56 views

Continuity of $F(x,y)=|x-y|$

Suppose that $F:\mathbb{R}^2\to \mathbb{R}$ defined by $F(x,y)=|x-y|$. Prove using $\epsilon-\delta$ that $F(x,y)$ is continuous. Let $(x_0,y_0)\in \mathbb{R}^2$. We have to show that for any $\...
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2answers
81 views

How to show that$\int_{0}^{\infty} \frac{sin(x)}{x}dx$ exists

How does one show that $\int_{0}^{\infty} \frac{\sin(x)}{x} \mathrm{d}x$ exists (i.e. does not equal $\infty$), with the most elementary methods possible?
2
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1answer
27 views

Confusion in finding left and right hand limits [duplicate]

Let $f:\mathbb R$→$\mathbb R$ defined as - $f(x)=0$, if $x$ is irrational or $x=0$ and $f(x)=1/q$, if $x=p/q$, $p\in$$\mathbb Z$ ,$q\in$$\mathbb N$, $(p,q)=1$. What are the points of continuity of $...
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23 views

Proof of Brouwer fixed point theorem using change of variable formula. [on hold]

Is there a proof of Brouwer fixed point theorem using change of variable formula for integration?
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0answers
32 views

Bounded convergence theorem for Riemann integrals

I will teach a analysis class to some olympiad students this month. The subject is the fundamental theorem of algebra. My approach will be to prove the following "modified" version of the bounded ...
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1answer
44 views

Nearest neighbor of an irrational number

I am confused in my thoughts about the irrational numbers in real line. My confusion is: If $x\in$$\mathbb R$$-\mathbb Q$ then for $\epsilon>0$ as small as you please, the element ($x+\epsilon$) ...
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18 views

Question on product measure: $\int_{[0,R]^2} g(x,y) df_1(x) \times df_2(y) = \int_{[0,R]} \left( \int_{[0,R]} g(x,y) df_1(x) \right) df_2(y)$ holds?

Suppose I have a real valued positive increasing functions $f_1(x), f_2(y)$. Then we know we can define Riemann-Stieltjes integral by defining measures $df_1(x)$ and $df_2(y)$. Let $g(x,y)$ be a ...
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0answers
16 views

On sharp bounds of some dyadic operators

I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. For example, the martingale transform $T_{\sigma}$ which ...
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2answers
31 views

$\displaystyle \lim\limits_{M\to\infty}\int_{0}^\infty (1+x^2)^{-s}\frac{\sin Mx}{x}dx$

Does the limit $\displaystyle \lim\limits_{M\to\infty}\int_{0}^\infty (1+x^2)^{-s}\frac{\sin Mx}{x}dx$ exist? Where $s>0$ be a fix real number. i.e. does the integral $\displaystyle \int_{0}^\...
2
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2answers
42 views

Inequality involving ArcTan

How to prove that for $x\in[0, +\infty]$ the following inequality is true: $$\arctan x\geq\frac{3 x}{1+2\sqrt{1+x^2}}?$$ I don't have idea from where to start, so any hint is welcome. Thanks in ...
2
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1answer
30 views

definition of derivative

Definition: A mapping $f:U\to \mathbb{R}^n$ from an open set $U\subset \mathbb{R}^m$ into $\mathbb{R}^n$ is differentiable at a point $a\in U$ if there is a linear mapping $A:\mathbb{R}^m\to \mathbb{R}...
2
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1answer
16 views

Exponential limit convergence for each $x$

I have $f_n(x)=\left( 1+\frac{-e^{-x}}{n} \right)^n$, what about the convergence to $f(x)=e^{-e^{-x}}$? is it true $\forall x?$ I say yes, but how can I show this? Is continuity of $f_n$ enough?
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1answer
30 views

Question involving continuity of function

Problem: Function $f$ is defined: $f(x)=x^2$ for $x\in \mathbb Q$ and $f(x)=x$ for irrational $x$. I have to check continuity of function. My work: Let $c\in \mathbb R\setminus \mathbb Q$. ...
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3answers
56 views

How to solve the limit of this sequence

$\lim\limits_{n \to \infty}(\frac{1}{3\cdot 8}+\dots+\frac{1}{6(2n-1)(3n+1)})$ I have tried to split the subset into telescopic series but got no result. I also have tried to use the squeeze theorem ...
2
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2answers
60 views

Closed form for an integral with log and power

Let $n \in \mathbb{N}$. We know that: $$\int_0^1 x^n \log(1-x) \, {\rm d}x = - \frac{\mathcal{H}_{n+1}}{n+1}$$ Now, let $m , n \in \mathbb{N}$. What can we say about the integral $$\int_0^1 x^n \...
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18 views

from rolle can we conclude the existence of a local extremal.

Let $f:[a,b]\longrightarrow \mathbb R$ continuous and derivable on $]a,b[$ s.t. $f(b)=f(a)$. Can I conclude the existence of a local extremum ? To me it looks obvious that yes, but I can't prove it. ...
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2answers
23 views

Find the supremum and infimum

I have a set $E = \{x: x^2-x-1 < 0 \}$ for which I need to find the infimum and supremum (and minimum and maximum if exists). I'm not sure how to do it but after some calculation I cam up with $Inf(...
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0answers
13 views

numerical integration asymptotic relation

Let $Q\subset R^n$ be a convex subset and $f\in C^2(Q)\;$ We set $x_s:=\int_Q xdx$,$\;\;\;Vol(Q):=\int_Q 1dx$ and $diam(Q)=sup||x-y||_2$ Prove the following asymptotic relationship: $...
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1answer
33 views

Complete a proof that $F(x,y)$ is contracting.

Can anyone fill in the dots in this proof? Let $D := [0,\frac{1}{2}]^2$. Show there is exactly one $(x,y)=(x^*,y^*)\in D$ such that \begin{align*} x &= \frac{x^3}{2} + y^4 + \frac{1}{4} \,, \...
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1answer
47 views

Measurable function and the Mean Value Theorem

Let $\,f:[a,b]\to \mathbb{R}\,$ be continuous on $[a,b]$ and derivable on $(a,b)$. By the mean value property, for all $\,x\in (a,b)\,$ there exists $\,\xi_x\in (a,x)\,$ such that $\,f(x)-f(a)=f'\left(...
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1answer
18 views

square of polynomial still harmonic? [on hold]

Let $P(z)=\sum_{i=0}^n a_i z^i$ be a polynomials on $\mathbb{C}[z]$ such that $a_i$ are real numbers. $|P(z)|^2$ is a harmonic function ?
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2answers
45 views

Theorem 2.17 from RCA Rudin

I understood the proof of points $(a)$ and $(c)$. But I can't understand the proof of $(b)$. It's obvious that every closed set is $\sigma$-compact. But how Rudin applies $(a)$ here? We have to show ...
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3answers
187 views

Derivative of the magnitude of a vector. Does it exist, or not?

I have a puzzling situation involving derivatives. I want to derivate: $$ \frac{d}{dx}| \mathbf F(x)| $$ This was actually something involving physics. Lets be 2-dimensional for simplicity. Let a ...
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0answers
20 views

Generalization of Strict Local Maxima

I try to generalize a strict local maximum to a local roof which can possibly be a flat area instead of just a single point. Below is my attempt: Let $f$ be a continuous real-valued function on $R^D$ ...
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0answers
36 views

How is the derivative exact if it's the standard part? [on hold]

I understand that no such infinitesimal value exists in our universe, so we round to the nearest real value when taking the derivative. Such as how $2x+\Delta x$ equals $2x$ when we take the standard ...
0
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1answer
37 views

Real analysis reference for statistician

I'm a undergraduate statistics student, I think that learn Real Analysis can be useful to me in some points, can anyone suggest a introductory book for self-study ? I'm already multivariate calculus, ...
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2answers
23 views

Closure of sets (specifically regarding the notation)

I'm new to sets and the notation is somewhat confusing to me. I just want to see if what I'm doing makes sense. For the following sets I need determine if it is open, closed, or neither. I also ...
3
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0answers
61 views

Are all important function spaces vector spaces?

EDIT: I definitely agree with Mike Miller that the question as written originally/below is too general. I guess what I am trying to ask is that "does everything an analyst could ever care about have ...