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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17 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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48 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
32 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 ...
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1answer
48 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
61 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
86 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
28 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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27 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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33 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
46 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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19 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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26 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
35 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
44 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
31 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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3answers
26 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
31 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
58 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
67 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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19 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
24 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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21 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
38 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} \,, \...
4
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1answer
54 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
20 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 ?
2
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2answers
47 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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194 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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35 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
46 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 ...
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1answer
39 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
25 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 ...
5
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1answer
51 views

For an arbitrary uncountable set of irrational numbers, can I always construct a sequence from them that converge in the rationals?

Suppose you have a set $S$ of uncountably many irrational numbers. Can you construct a sequence of $S$ that converges to a rational number? What I have tried: Since $S$ is uncountable, the inf of ...
3
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1answer
53 views

Real Analysis, Folland problem 2.2.16 Integration of Nonnegative functions

If $f\in L^+$ and $\int f < \infty$, for every $\epsilon > 0$ there exists $E\in M$ such that $\mu(E) < \infty$ and $\int_E f > (\int f) - \epsilon$. Attempted proof - Let $f\in L^+$ and ...
3
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1answer
35 views

Boundary and Interior of set $\{-3,2,5\}$

I'm trying to see if I'm correctly understanding and applying the definition for interior and boundary points. Interior point: A point x in R is an interior point of S if there exists a ...
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2answers
54 views

Real Analysis, Folland Problem 2.2.14 Integration of Nonnegative functions

Problem 2.2.14 - If $f\in L^{+}$, let $\lambda(E) = \int_{E}f d\mu$ for $E\in M$. Then $\lambda$ is a measure on $M$, and for any $g\in L^{+}$, $\int g d\lambda = \int f g d\mu$.(First suppose that $g$...
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2answers
49 views

Dual result of Fatou lemma

If $\{f_n\}\subset L^+$, $f\in L^+$ such that $\{f_n\}$ is dominated by $f$ where $\int f < \infty$ then $$\limsup\int f_n\leq \int \limsup f_n$$ Attempted proof - Consider the sequence $\{f - ...
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20 views

How am I to understand this notation with regards to bdS and the int S? $S=\bigcap_{n=1}^{\infty}\left(-\infty,7+\frac{1}{n}\right]$

I'm trying to find the largest $\epsilon$ such that the neighborhood centered at $x$ of radius $\epsilon$ is contained in $S$. That part I think I can do, but I just don't know understand the below ...
0
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3answers
59 views

How to find a parametrization for a torus?

I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$ where $0<r<R$. I know I need to compute the metric tensor and ...
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3answers
77 views

Question involving limit

How to find the following limit: $$\lim_\limits{n\to\infty} (ne\sqrt[n]{\ln{(1+e^n)}-n}-n)$$ Thanks in advance!
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1answer
29 views

Real Analysis, Folland Corollary 2.19 Integration of Nonnegative functions

Corollary 2.19 - If $\{f_n\}\subset L^+$, $f\in L^+$, and $f_n\rightarrow f$ a.e., then $\int f \leq \liminf\int f_n$. Proof - We have that $\{f_n\}\subset L^+$, $f\in L^+$ and $f_n\rightarrow f$ a....
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3answers
104 views

Help, Where is wrong when I do same complex integration using two different contours

everyone! please give few hit. I want take the integral $$I=\int_{0}^{\infty}{\frac {dx}{ \sqrt{x}(1+{x}^{2})}} $$ by using the Residue Theorem. I choice two contours in complex plane with $z=r e^{i\...
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2answers
40 views

$\lim _{ x\rightarrow 0 }{ \left( \csc ^{ 2 } x-{ x }^{ -2 } \right) } $ [closed]

$\lim _{ x\rightarrow 0 }{ \left( \csc ^{ 2 } x-{ x }^{ -2 } \right) } $. It is of the form ($\infty$-$\infty$).
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3answers
50 views

Uniform convergence of $(1 − x^{n+1})/(1 − x)$ [closed]

Prove that the sequence $(1 − x^{n+1})/(1 − x)$ converges uniformly to $1/(1 − x)$ on each interval of the form $[−r, r]$ with $r < 1,$ but it does not converge uniformly on $(−1, 1).$
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37 views

Baby Rudin existence of smooth function for every closed set as it's zero set

Problem 21 of chapter 5 asks whether a function can be found for every closed set $F$ in $R$, with it's zero set precisely $F$ having derivatives of all orders. The following solution problem is ...
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15 views

Equivalent definition of Lebesgue measurability in terms of additivity?

When introducing measurability, we noted that we wanted the following property to hold for $A, B \in \mathcal{P}(\mathbb{R})$ $m(A \cup B) = m(A)+m(B)$ (additivity) We then defined a set A to be ...
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4answers
41 views

Correctness of proof that every neighborhood is an open set.

Rudin makes the following definitions: (a) A neighborhood of p is a set $N_r(p)$ consisting of all $q$ such that $d(p, q) < r$, for some $r > 0$. (b) $E$ is open if every point of $E$ is an ...
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1answer
42 views

Produce a sequence $(g_n):g_n(x)\ge 0$ and $\lim g_n(x)\neq 0$ but $\int_{0}^{1} g_n\to 0$

Produce a sequence $(g_n):g_n(x)\ge 0,\,\forall x\in [0,1],\,\forall n\in\Bbb N$ and $\lim g_n(x)\neq 0,\,\forall x\in [0,1]$ but $\int_{0}^{1} g_n\to 0$ Im in need to clarify that Im talking of ...
0
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3answers
84 views

The sequence defined by $x_{n}=\frac{x_{n-1}+x_{n-2}}{2}.$

Let $x_{1}=0,x_{2}=1$ and for $n\geq3,$ define $x_{n}=\frac{x_{n-1}+x_{n-2}}{2}.$ Which of the following is/are true? $1.\{x_{n}\}$ is a monotone sequence. $2. \lim_{n\to\infty} x_{n}=\frac{1}{2}.$ ...
2
votes
1answer
54 views

$m_*(E)=m^*(E)\iff E$ Lebesgue measurable

Let $E\subset [a,b]$. Show that $E$ is Lebesgue measurable if and only if the Lebesgue outer measure of $E$ is equal to the Lebesgue inner measure of $E$. I have seen the proof for this above ...