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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Can we find two measures $\nu$, $\mu$ which $\nu\ll\mu$ and $\mu$ is $\sigma$-finite while $\nu$ is not $\sigma$-fintie?

Can we find two measures $\nu$, $\mu$ which $\nu\ll\mu$ and $\mu$ is $\sigma$-finite while $\nu$ is not $\sigma$-fintie? I want to justify the Radon-Nikodym theorem but couldn't find an example.
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8 views

A Borel-Cantelli lemma exercise.

Suppose ${A_n}$ is a sequence of events. If $P(A_n)\to 1$ as $n\to\infty$,prove there exists a subsequence ${n_k}$ tending to infinity such that $$P(\cap_kA_{n_k})>0$$ The textbook gives a hint ...
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159 views

Integral inequality: $\def\intd{\,\mathrm d}\int_a^b(f'(x))^2\intd x-2\big(f(a)+f(b)\big)^2\geq\frac8{(b-a)^2}\int_a^b(f(x))^2\intd x$

I have a problem which I think is wrong. Let $f: [a,b] \to \mathbb{R}$ be a differentiable function with $f'$ continuous such that $$\int_a^b f(x) \intd x = f\left(\frac{a+b}{2}\right) = 0$$ ...
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31 views

radius of convergence for the series $\sum_{n=1}^{\infty}(2^n+3^n+4^n) $ $x^{n}$

How do we solve it to get the radius of convergence radius of convergence for the series $\sum_{n=1}^{\infty}(2^n+3^n+4^n) $ $x^{n}$
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13 views

Exercise 3.3.8 from Understanding Analysis by Stephen Abbott

Motivation: trying to prove that if $K \subseteq \mathbb{R}$ is compact (and thus, by the Heine-Borel theorem, closed and bounded), then this implies that any open cover for $K$ has a finite subcover. ...
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1answer
24 views

$f\in C(\mathbb{R})$. What does it mean?

$f\in C(\mathbb{R})$. What does it mean? My guess is "Differentiable on $\mathbb{R}$" but I'm not sure.. Thanks.
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1answer
10 views

Continuity of Thomae's Function at Irrationals

It is well known that the Thomae's Function is continuous at irrationals and discontinuous at rationals. I used the following definition to prove discontinuity at rationals: A function $f\colon ...
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0answers
9 views

What is the nature of this one dimensional function?

Let $\mathcal{S}$ be a 2-D convex set whose elements can be represented as $(x,y)\in\mathcal{S}$. Let $p_L$ and $p_U$ be two real constants such that $p_L\leq p_U$. For $p\in[p_L,p_U]$, I define the ...
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1answer
27 views

Prove that $|f ''|\ge 4$

Let $f(x)\in C^2:[0,1]\rightarrow\mathbb{R}$ satisfy $f(0)=0,f(1)=1,f'(0)=f'(1)=0$, prove that: $$\max_{x\in[0,1]}|f''(x)|\ge4$$ By using Taylor series I can prove that ...
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1answer
13 views

Proof check - equivalence classes are intervals

We define $\sim$ for a nonempty subset $X\subseteq\mathbb{R}$ by: $x\sim y$ if $\lbrack\min\lbrace x,y\rbrace,\max\lbrace x,y\rbrace\rbrack\subseteq X.$ This is an equivalence relation on $X.$ I want ...
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7 views

Non-convex constraint made cost

Consider the non-convex optimization problem $$ \min_{x \in X} \ f(x) \quad \text{s.t.:} \ \ g(x) \leq 0, \ h(x) = 0 $$ where $X \subset \mathbb{R}^{2n}$ is compact and convex, $f$ and $g$ are ...
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2answers
27 views

Relation between different forms of Cauchy-Kovalevskaya Theorem

Consider the following form of Cauchy-Kovalevskaya (CK) theorem for a system of PDE: For a given system of PDE with $p$ unknown functions $u^{(1)}, \dots, u^{(p)}$ and variables $x, y_1, \dots, ...
3
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1answer
22 views

$\int f = \lim\int f$ but $\int_{E}f\neq\lim\int_{E} f_{n}$

This is exercise 2.13 in Folland's Real Analysis textbook Let $(X, \mathcal{M})$ be a measurable space. Suppose $\{f_{n}\}\subset L^{+}$, $f_{n}\to f$ pointwise, and $\int f=\lim\int ...
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1answer
23 views

Norm of functional associated to vector $p$-norm [duplicate]

I read that the norm of a linear functional $f:V\to K$, with $K=\mathbb{R}\lor K=\mathbb{C}$, associated to the $p$-norm $\|x\|=(\sum_{i=1}^n|x_i|^p)^{\frac{1}{p}}$, for $p>1$, is ...
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40 views

A identity relating a infinite series and a definite integral

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
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2answers
30 views

If $a_n\to0$, there exists $\pm$ such that $\sum\limits_n\pm a_n$ converges [duplicate]

Our Analysis I lecturer in his last lecture for the course gave us a problem to think about. I've been thinking about it for a while and has been bothering me for some time. It looks like a ...
0
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1answer
412 views

Min, Max, Infimum, and Supremum

I have tried to answer the following question but I’m not sure if I’m on the right track… please give me any feedback!! Find the min, max, infimum, and supremum for each of the following. ...
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20 views

How many sequence of functions are there to converge pointwise to a given function on $E\subseteq \mathbb R$?

yesterday night, I was studying sequence of functions in $\mathbb R$ and then this question came to mind. When a sequence of real valued function is given, we can find out it's pointwise limit ...
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26 views

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$ My attempt: Set $f_n(x) = \sin(nx)$. Argue by contradiction, suppose there exists a Cauchy ...
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2answers
23 views

sequence: use of stirling formula

I want to use the Sterling formula which says that: $lim_{n \rightarrow \infty}\dfrac{n!}{\sqrt{2*\pi}n^{n+1/2}*e^{-n}}=1$ I want to use it to show that $\lim_{n \rightarrow ...
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6 views

Exterior measure of a subset $A \subset \mathbb R_n$ equals the measure of a$G_{\delta}$

Let $A \subset \mathbb R^n$, prove that there is $H$: $A \subset H$, with $H$ a $G_{\delta}$ set such that $|A|_e=|H|$. The definition of $|A|_e$ is $|A|_e=\inf\{m(U): A \subset U\}$ where the ...
4
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1answer
36 views

Mixed partial derivatives are different

Let $f: \Bbb R^2 \to \Bbb R$ be defined as $$f(x) = \left\{ \begin{matrix} x_1^2 \operatorname{arctan} \left( \frac{x_2}{x_1} \right) - x_2^2 \operatorname{arctan} \left( \frac{x_1}{x_2} \right), ...
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1answer
20 views

geometric interpretation of the norm: $\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$

Let $p:\mathbb R^2 \to \mathbb R$ be a norm so that $$\|\vec x\|={(|x_1|+|x_2|)\over 3}+{2\max(|x_1|,|x_2|)\over 3}$$= $${{\|\vec x\|_1\over 3}}+{2\|\vec x\|_\infty\over 3}$$ The thing is that I need ...
0
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1answer
42 views

A problem about Borel-Cantelli lemma

I am doing exercise in a textbook and I got confused with these two problems: Fisrt, in problem 21,I noticed that "there is a subsequence $\{n_k\} $ tending to infinity s.t. ...
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34 views

Seeking for concrete examples of extended Radon-Nikodym theorem

Extended Radon-Nikodym theorem: measurable space:$(\Omega,\mathcal{A})$ $\mu $ is a $\sigma$-finite measure. $\nu$ is a measure(not $\sigma$-finite) which has the property "$\nu ...
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4answers
671 views

How to prove that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ $H_n$ denotes the harmonic numbers.
4
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1answer
170 views

Replacing the “if $x ≤ y$, then $x + z ≤ y + z$” axiom in Reals.

How can I prove that we cannot (or maybe can) replace preservation of order under addition i.e. "If $x \leq y$, then $x + z \leq y + z$ with "if $0<x$ and $0<y$ , then $0<x+y$" in axioms ...
1
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1answer
19 views

Lipschitz constants of projections

Consider two compact sets $A, B \subset \mathbb{R}^n$. Assume that the projection mappings $P_A: \mathbb{R}^n \rightarrow A$, $P_B : \mathbb{R}^n \rightarrow B$ have Lipschitz constant $1$ and $L$, ...
9
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1answer
48 views

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$. Proof: suppose $T:L^1 \rightarrow L^\infty$ continuous and onto. $L^1$ is separable, let $\{f_n\}$ be a countable dense ...
2
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0answers
39 views

Is this a valid proof? (epsilon delta)

I'm asked to prove using an $\epsilon - \delta$ - argument that $\displaystyle \lim_{(x,y) \to (0,0)} \frac { 4x^3 + 8xy^2}{3\sqrt[3]{(x^2 + y^2)^4}} = 0 $. Here is the proof I came up with: Let ...
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1answer
23 views

exercise on pointwise convergence of an (easy) function.

Exercise 6.2.5. Taken from understanding analysis of Stephen Abbott For each n $\in N$, define $f_n on \ R$ by $$f_n(x) = \begin{cases} 1, & \mbox{if} \ |x| \ge 1/n \\ n|x|, & \mbox{if} \ ...
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0answers
14 views

Equivalent optimization problems?

I am wondering if the set of optimizers of the problem $$ \min_{x \in X} \ f(x) \quad \text{subject to: } g(x) \leq 0, \ h(x) = 1 $$ is the same of the one of $$ \min_{x \in X} \ f(x) + h(x) \quad ...
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22 views

is the upper limit projection Borel

Let $M$ space metric compact, $\pi:M\times\mathbb{R}^k\rightarrow M$ projection such that $\pi(x,y)=x$. Let $f_n:M\times\mathbb{R}^k\rightarrow \mathbb{R}$ continuous and ...
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1answer
28 views

Projection of a set $G_\delta$

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps $G_\delta$ sets to Borel sets? i.e. If $A=\cap_n^\infty A_n$ with $A_n$ open sets, then $\pi(A)$ is ...
0
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3answers
196 views

Invertibility of $I-A$ if the spectral radius of the operator $A$ is less than $1$

I want an explication of the following fact: If the spectral radius of a bounded operator $A$ on a Banach space is less than one, then $I - A$ is invertible.
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2answers
30 views

Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, ..., $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p'}(E)$, ...
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Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function
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27 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
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1answer
39 views

Calculating multi-variable limit.

I am struggling to find a way to approach this limit $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2y+x^2y^3)}{x^2+y^2}$$ I would greatly appriciate if You could explain to me how to solve it or at least show ...
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4answers
2k views

Is the function $ f(x,y)=xy/(x^{2}+y^{2})$ where f(0,0) is defined to be 0 continuous?

Is the function $ f(x,y)=xy/ (x^{2}+y^{2})$ where $f(0,0)$ is defined to be $0$ continuous? I don't think it is and I am trying to either show this by the definition or by showing that maybe a close ...
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1answer
40 views

A problem on verify conditional expectation

Suppose X and Y are independent.Let $\varphi $ be a function with $E(|\varphi(X,Y)|)< \infty$ and let $g(x)=E(\varphi(x,Y))$.The conclusion is $E(\varphi(X,Y)|X)=g(X)$ So the first step is to ...
0
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1answer
19 views

A problem about indefinite integral in measure theory

tirple$(\Omega,\mathcal{A},P)$ Suppose $\xi$ is a random variable.Indefinite integral$$\varphi(B)=\int_B\xi\mathbb{d}P \quad\forall B\in\mathcal{A}$$ I saw in a textbook: If $E(\xi)$ exists(not ...
19
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3answers
5k views

Are calculus and real analysis the same thing?

I guess this may seem stupid, but how calculus and real analysis are different from and related to each other? I tend to think they are the same because all I know is that the objects of both are ...
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10 views

Is there a method to find the equation of a parabolic branch?

Does it exist a method to find the the equation of a parabolic branch? Assume we have $f(x)\rightarrow +\infty$ when $x\rightarrow +\infty$ and $\frac{f(x)}{x}\rightarrow +\infty$ when $x\rightarrow ...
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0answers
47 views

Problem book for calculus .

I'm a high school student and have followed Apostol for learning calculus. What I found extremely interesting is the part on sequence and series along with the differential calculus part. Can anyone ...
3
votes
2answers
97 views

Harmonic number divided by n [duplicate]

How do I prove that $\dfrac{H_n}{n}$ (where $H_n$ is a harmonic number) converges to $0$, as $n \to \infty$?
0
votes
1answer
28 views

Proving differentiability using Caratheodory's Lemma

Let $I$ be an open interval and let $c\in I$. Let $f:I\rightarrow\mathbb{R}$ be continuous and define $g:I\rightarrow\mathbb{R}$ by $g(x)=\left|f(x)\right|$. Prove that if $g$ is ...
1
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0answers
40 views

Is an inner product continuous?

It is an easy question, but i want to make it clear :) Let $(V,\langle -,- \rangle)$ be an inner product space over $\mathbb{K}$. Then, is the inner product $\langle -,- \rangle:V\times V\rightarrow ...
4
votes
1answer
79 views

Sequence with equidistant terms

Consider the sequence $(u_{n})_{n \in \mathbb{N}}$ given by : $$ u_{0} \in \mathbb{Z} \quad \mathrm{and} \quad \forall n \in \mathbb{N}, \, \vert u_{n+1}-u_{n} \vert = 1.$$ Is the sequence ...
1
vote
1answer
30 views

Radius of convergence of power series $\sum_{n=1}^{\infty}{\frac{\sin n!}{n!}} {x^n}$

The power series $\displaystyle\sum_{n=1}^{\infty}{\frac{\sin n!}{n!}} {x^n}$ has radius of convergence $R$, then $R\geq1$ $R\geq e$ $R\geq2e$ All are correct I wanted to know how will get the ...