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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1answer
8 views

Show that this integral is finite $\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx$

Let $p > -1$ and $r \in \mathbb{N}$, show that $$\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx = \int_0^\infty x^p (\ln x)^r e^{-x} dx$$ and that this integral is finite. To ...
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1answer
9 views

The geometry meaning of Riemann–Stieltjes integral

Maybe my question seems so strange but I want to know what is the geometry meaning of Riemann stieltjes integral ??
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14 views

Rational,and irrational number between any 2 real numbers.How to prove?

If $x,y$ are 2 real numbers such that $x < y$ how to to prove there is an $r$, belongs to the set of rational numbers, and a $i$, belongs to the set of irrational numbers and hence many more ...
1
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1answer
25 views

Simple calculus series question; convergence of $\sum_{j = -\infty}^{\infty} \frac{1}{z - j}$

So I am having a brain fart and I cannot rigorous write (in my head) why the series in the question does not converge. I know it has to do with the harmonic series. Is it because if $|z| \to \infty$, ...
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2answers
25 views

How fast does this function converge to zero?

Consider the function given by $$f:(0,\infty)\rightarrow \mathbb{R}, t\mapsto \int\limits_{-\delta}^{\delta}x^{2k}\frac{1}{\sqrt{4\pi t}}e^{-\frac{x^2}{4t}}dx,$$ where $k\in\mathbb{N}$ and $\delta ...
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0answers
15 views

Almost everywhere differentiability

Suppose $f: \mathbb{R} \to \mathbb{R}$ is increasing and $g = f$ almost everywhere with respect to Lebesgue measure (a.e.). Suppose $g'$ exists a.e. Does it follow that $g' = f'$ a.e.?
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1answer
14 views

Does it hold $\sigma(X_1,\ldots,X_n)=\sigma(X_n-X_1,\ldots,X_n-X_n)$?

Let $(\Omega,\mathcal{A})$ and $(\Omega',\mathcal{A}')$ be measurable spaces $X_1,\ldots,X_n$ be measurable with respect to $\mathcal{A}$-$\mathcal{A}'$ $Y_m:=X_n-X_m$ for $1\le m\le n$ I'm ...
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1answer
17 views

if $g(t)=f(\frac {\cos (t)}t,\frac {\sin (t)}t)$ is a monotonic increasing function for $t>0$ then $\nabla f(0,0)=(0,0)$

Let $f:\mathbb R^2 \rightarrow \mathbb R $ be in $C^1$. suppose that $g(t)=f(\frac {\cos (t)}t,\frac {\sin (t)}t)$ is a monotonic increasing function for $t>0$. prove that $\nabla f(0,0)=(0,0)$. I ...
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1answer
15 views

weak-star convergence to Dirac Delta function

Let $Y=C_c((-1,1))$. Let $f_j=2j\chi_{(-1/j,1/j)}$, $f_j:(-1,1)\to\mathbb{R}$. Let $\Lambda_j:Y\to\mathbb{R}$ defined by $\Lambda_j(g)=\int f_jg$ and $\Lambda: Y\to\mathbb{R}$ defined by ...
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0answers
10 views

Othonormal basis for $L^2$ space on square

Can one describe an orthonormal basis for $L^2(\gamma,ds)$ where $\gamma$ is the square with vertices at $(1,1),(-1,1),(1,-1),(-1,-1)$ and $ds$ is the arc length. To be more precise can we express ...
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2answers
14 views

finding $C^1$ path on an open and path connected set.

Given an open and path connected set $U\subseteq \mathbb R^n$, is there a way to find a $C^1$ path between every $a,b\in U$? If so, is there a general proof of existence of such path?
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0answers
9 views

On an algebraic manipulation of a double summation used to obtain the kth ordinate of the periodogram.

I am following Introduction to statistical time series by Fuller. I am having some problems with what I think is an algebraic manipulation of the double summations in the line where the mouse pointer ...
0
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1answer
20 views

Prove linear mapping belongs to $L: l^{2}(\mathbb{R})^{\star}$ and find operator norm

Prove that the linear mapping $L: l^{2}(\mathbb{R})\rightarrow\mathbb{R}$ defined by $$L({x_n})=\sum_{n=1}^{\infty}{x_n}/(2^{n/2})$$ belongs to $L: l^{2}(\mathbb{R})^{\star}$. What is its operator ...
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0answers
25 views

Application of Sharkovskii's theorem

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous, $n \geq 3$ and $x_1< \dots < x_n$ so that $f(x_i)=x_{i+1}$ for all $i=1,\dots n-1$ and $f(x_n)=x_1$. In order to apply the theorem I have ...
1
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1answer
46 views

Understanding the proof of the Arzelà–Ascoli theorem from wikipedia

This is the statement and the proof from wikipedia. I have highlighted the part of the proof that i have questions with. Let I = [a, b] ⊂ R be a closed and bounded interval. If F is an infinite set ...
3
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3answers
55 views

How do I rigorously show that $f(x, y) = \frac{x}{2|x|\sqrt{|x|+|y|}}$ is continuous when $x, y \neq 0$?

For the function $f : \mathbb{R}^2 \to \mathbb{R}$ to be continuous, I need to show that for some given $\epsilon > 0$, there exists a $\delta > 0$ so that if $||z - z'|| < \delta$, then ...
2
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0answers
39 views

reference required for a well-known result

Consider the following classic problem: Given a 2-dimensional cube (a square) $[0,1]^2$, one of the minimum (area) simplex covering the 2-cube is bounded by $x_1 =0 , x_2=0, x_1+x_2 =2$. For ...
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2answers
24 views

Radius of convergence problem.

Consider the power series $\sum a_nx^n$ with radius of convergence R . a) Prove that if all the coefficients $a_n$ are integers and infinitely many of them are non zero then , R is less then equal to ...
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1answer
27 views

Homeomorpic spaces

We define the two following sets: $ E_1 =\{ (L,v) \in \mathbb{R}P \times \mathbb{R}^{1} \mid v \in L \ or \ v=0\} $ and $A=([0,1] \times \mathbb{R})/\sim $ . Where $\sim$ is the equivalence ...
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0answers
26 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if variables are gaussian?

Is the Gaussian random variable with density given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{x^2}{2 \sigma^2}}$$ the unique RV such that the joint pdf of $N > 1$ independent and ...
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0answers
17 views

Proving that a function has compact support

How can I argue/show in a proper way that if $f \in L^2$ has compact support and $\phi \in L^2$ with $\{T_k\phi\}_{k\in\mathbb{Z}}$ being an orthonormal system where $T_k$ is the translation operator, ...
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0answers
10 views

Judging whether a function is integrable

Here, f(x,y) is defined on [-1,1] x [-1,1]. I tried to calculate the integration of absolute value of f on the domain, using Tonelli's theorem. But the function is too complicated for me to ...
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0answers
11 views

Proving differentiability and non-differentiability for multivariable functions

I've been stuck on this problem for a while now. I looked at my notes, and if for a function, all the partial derivatives of its matrix of partial derivatives exist and are continuous in a ...
0
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1answer
25 views

finding $\lim_{h\to 0 }\int \vert f(x+h)-f(x)\vert ^pdx$ when $f\in L^p$

Suppose $f\in L^p$, $p\in [1,\infty)$. My naive attempt is $\lim_{h\to 0}\int \vert f(x+h)-f(x)\vert ^pdx=\lim_{h\to 0}\int \vert \frac{f(x+h)-f(x)}{h}\vert ^ph^p=\int \vert f'(x)\vert ^ph^p=0$, but I ...
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2answers
30 views

Can be $f_n:[0,1]\rightarrow\mathbb{R},\:f_{_n}(x)=x^n \cdot\ e^x$ uniform convergence?

We have $f_n:[0,1]\rightarrow\mathbb{R},\:f_{_n}(x)=x^n \cdot\ e^x$. I don't know how we can find the pointwise convergence...This sequence can be a uniform convergence? and explain your argument.
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0answers
11 views

Definition of rough path

There are many books and notes on the rough path theory. A rough path is defined as an ordered pair $(X, \mathbb X)$, where $X$ is a path mapping from $[0, T]$ to some Banach space $V$ and $\mathbb X: ...
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1answer
32 views

Power Series Problem.

Suppose $\sum a_nx^n$ has a finite radius of convergence, say $R$, and $a_n\ge 0$ for all $n$, show that if the series converges at $R$ then it also converges at $-R$ . What I did: Applying the root ...
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0answers
7 views

Non periodic Fourier Series Point Convergence

If $f$ is a real-valued non-periodic continuous function that is differentiable at the point $x_0$, is it true that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
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1answer
20 views

Prob. 4, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $(e_n)$ be an orthonormal sequence in an inner product space $X$. Then, for every $x \in X$, we have $$ \sum_{n=1}^\infty \vert \langle x, e_n \rangle \vert^2 \ \leq \ \Vert x \Vert^2.$$ Now ...
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1answer
18 views

Prob. 3, Sec. 3.4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to derive the Schwarz inequality?

Let $\left( e_n \right)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X$, we have $$ \sum_{n=1}^\infty \left\vert \langle x, e_n \rangle \right\vert^2 \ \leq \ ...
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0answers
12 views

Limits of Taylor POlynomials over $k$-tuples?

Let $f \in \mathscr{C}^{(m)}(E),$ where $E$ is an open subset of $R^{n}$. Fix $\textbf{a}$ $\in E$, and suppose $\textbf{x}$ $\in R^{n}$ is so close to $\textbf{0}$ that the points \begin{equation*} ...
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0answers
11 views

weak solution to one dimension conservation law

Suppose $u:\Bbb{R}\times[0,\infty)\to\Bbb{R}$ is a continuous function such that for all $v\in C_c^\infty(\Bbb{R}\times[0,\infty))$ $$ \int^{\infty}_0 \int^{+\infty}_{-\infty} \Big(u(x,t) ...
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0answers
17 views

An example in which the Fubini theorem is inapplicable

This is example 8.9(a) in Rudin's Real and Complex Analysis, (alternatively, exercise 10.2 in Rudin's Principles of Mathematical Analysis). Let $X$ and $Y$ be the closed unit interval $[0,1]$, let ...
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2answers
52 views

The norm $\|f_n-f\|_{L^1} \to 0$ but $f_n \not\to f$

A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12: Show that there are $f \in L^1(\mathbb{R}^d)$ and a sequence $\{f_n\}$ with $f_n \in ...
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0answers
20 views

Prove integral belongs to dual space and find operator norm [on hold]

Prove that the linear mapping $L: C([0,1])\to \mathbb{R}$ defined by $$L(f)=\int_{0}^{1}f(x)dx$$ belongs to $C([0,1])^{\star}$. What is its operator norm? ($C([0,1])^{\star}$ is the dual ...
0
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1answer
12 views

Proving uniform approximation by polynomials when sets are not compact

Here are two problems of the same flavor (and hence I posted them simultaneously) based on the Stone-Weierstrass Approximation Theorem. Let $f$ be continuous on $[1,\infty)$ with ...
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2answers
71 views

Proving that $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.

Problem. Show that a bounded function $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable if and only if for every $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that for any partition $ ...
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0answers
27 views

Area of a surface of revolution about the y-axis-

I'm trying to find the area of a surface of revolution generated by the curves $$y=x^3,\quad x=1,\quad x=2, \quad\rm{around} \quad y=-1 $$ \begin{array}{lcl} A &=& 2\pi \int_1^2 {(y + 1)\sqrt ...
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0answers
16 views

For which $(x_1,x_2)$ is this a solution to the minimal surface equation?

Let $u(x_1,x_2):=arcosh(\sqrt{x_1^2+x^2})$ then I want to find out for which $(x_1,x_2)$ this is a solution to the minimal surface equation in two dimensions that you can find for example here. ...
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1answer
26 views

Show space C([0,1]) with norm integral is a Banach space [duplicate]

Is the space C([0,1]) with the norm integral from 0 to 1 of |f(t)|dt a Banach space?
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1answer
32 views

Connected Sets Examples

(a) Give an example of a connected set $A \subset \Bbb R^n$ such that $\Bbb R^n\setminus A$ is not connected. (b) Give an example of a compact set $K \subset \Bbb R^n$ which is not connected. So far ...
3
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2answers
71 views

Equality of a quadratic function

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ an arbitrary function and $g: \mathbb{R}\rightarrow \mathbb{R} $ a quadratic function with the following property: For any $m$ and $n$ the equation ...
0
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1answer
74 views

To prove $\sin(x) > x - \frac{x^3}{6}$ is strictly increasing [on hold]

How do I prove this is strictly increasing? $ \sin(x) > x - \frac{x^3}{6} $ where $ x>0 $ What I did so far, I first rearrange the inequality, Let $ f(x)=\sin(x) - x + \frac{x^3}{6} ...
0
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0answers
21 views

Creating a sequence convergent to zero with special characteristic

Let $\{a_k\}$ and $\{b_k\}$ be positive sequences in $\mathbb{R}$ that both converge to zero. Can we choose $\{c_k\}$ such that it converges to zero and $$ 0<\lim_{k \to \infty} \frac{a_k}{c_k} = ...
0
votes
1answer
24 views

Problems on Divergence theorem

I am struggling in the following problem: $ S \subset R^3$ is a region in divergence theorem. $\vec{n}$ is outward normal to the surface of $S$. Then, what does $div \vec{F}=0$ mean in the ...
2
votes
1answer
32 views

If E is measurable, then $\delta E$ is measurable.

Problem: If $\delta =(\delta_1,\delta_2,\cdots,\delta_d)$ is a d-tuple of positive numbers $\delta_i>0$, and $E$ is a subset of $\mathbf{R^d}$, we define $\delta E$ by $\delta E = ...
2
votes
1answer
52 views

How to prove that $|a_{1}+2a_{2}+…+na_{n}| \leq 1.$

Let $$f(x)=\sum_{k=1} ^{n}a_{k}\sin(kx)$$ where $n \in \mathbb{Z^+}$ and $a_{k} \in \mathbb{R}$ for each $k=1,…,n.$ Suppose that $ \vert f(x)\vert \leq \vert \sin(x) \vert $ for every $x.$ Prove ...
0
votes
1answer
21 views

Integration of step functions

I've managed parts (a) and (b) fairly easily, but c is causing me a real headache. I've seen the Cauchy-Schwartz inequality before, but I've hit a roadblock because I've no idea whether or not I can ...
0
votes
1answer
30 views

What is $\sum_{i,k=1}^n x_k^2x_i^2$?

I stumbled over the double sum $$\sum_{i,k=1}^n x_k^2x_i^2$$ and was wondering whether this is anything that can be expressed in terms of the euclidean norm? Maybe it is even $||x||^4$ but I am not ...
0
votes
1answer
27 views

Area of surface [duplicate]

What is the area of the region that is bounded by the curve $$\vec{R}(t)=(\cos^3t, \sin^3t), 0\leq t<2\pi?$$ I have no idea how to start here or what i have to use.