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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2
votes
2answers
45 views

Part of proof to show Lebesgue-lebesgue measurable

I want to prove the following: Suppose $E$ is a subset of $\Bbb R$, let $\gamma(E)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in E\}$. If $E\in \Bbb B$ (Borel/Lebesgue measurable set), show that ...
2
votes
0answers
15 views

Approximate an integrable function using a simple function (Proving existance)

Let $f \in L^1(\mathbb{R})$, and let $\epsilon > 0$. Show that exists simple function $g=\sum_{k=1}^{n}c_k 1_{A_k}$, such that, $$\int_\mathbb{R} |f(x)-g(x)|dx \leq \epsilon$$,and such that $n \in ...
-1
votes
1answer
14 views

A question about the real line and the Dirichlet function.

Though the graph of the Dirichlet function is non-drawable, I think if we have to draw it in some informal way then it will be two complete lines (instead of isolated points). Here's my reasoning: ...
0
votes
0answers
9 views

Infinite Sum of Incomplete Gamma (with “z” varying)

I am trying to see whether the following series can be "reduced" or re-written in terms of well-known functions. This seems different from other questions on here in that the summation is going ...
1
vote
1answer
50 views

Prove $(1+x)^p+(1-x)^p \ge 2(1+x^p)$ for $0\le x\le1$ and real number $p\ge2$.

I don't know how to prove the following questions: If $p\ge2$ is real, then $$ (1+x)^p+(1-x)^p \ge 2(1+x^p) \quad \text{for } 0\le x\le1; $$ if $1\le p<2$, then opposite direction of the inequality ...
1
vote
0answers
46 views

Show that $\iint_{X \times Y}\varphi(x)k(x,y)\psi(y) d(\mu \times \nu)=\int_Y \Big[\int_X\varphi(x)k(x,y)d\mu \Big] \psi(y) d\nu$

Let $k(x,y)$ be a bounded Borel measurable function on $X \times Y$ and let $\mu$ and $\nu$ be Radon measure on $X$ and $Y$ i. Show that $\iint_{X \times Y}\varphi(x)k(x,y)\psi(y) d(\mu \times ...
0
votes
1answer
49 views

A simple proof of the fact that $\int_0^{+\infty} \cos(x)/\sqrt{x} \text{d}x \neq 0$

When doing an exercise, I found that a sequence $(u_n)$ satisfies the following $$ u_n \underset{n\to + \infty}{\sim} \frac{1}{n^{\alpha/2}} \int_0^{n^\alpha} \frac{\cos(x)}{\sqrt{x}} \text{d}x, $$ ...
0
votes
1answer
50 views

How to solve this functional equation: $f(1-f(x))=1-x^{9}, f(1)=0$

I have managed to guess one solution of this function : $f(x)=1-x^{3}$, but I have no idea how to prove it unique, or get other solutions. If this is not solvable, how can you prove this function ...
0
votes
0answers
30 views

measure and integration theory [on hold]

Assume that a function $f:I\longrightarrow \mathbb{R}$ defined on a closed subinterval $I=[a,b]$ of $\mathbb{R}$ satisfies a Lipschitz condition. That is, assume that there exists a constant ...
0
votes
1answer
23 views

A question about Linear transformations on matrices

We know that if $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is a linear transformation, then there exists a matrix $A\in \mathbb{R}^{m\times n}$ such that $T(x)=Ax$, for every $x\in\mathbb{R}^n$. Now, if ...
1
vote
2answers
32 views

Helping understand line integral $\int_{K,+}{(x+y)}dx+(y-x)dy$

I have a huge problem with understanding line integrals and would be much obliged for your help! We have: $$\int_{K,+}{(x+y)}dx+(y-x)dy$$ and the following parameterization: ...
1
vote
2answers
51 views

Stuck with homework (limit exercise) - squeeze theorem

Given function $f:\mathbb R→\mathbb R$ for which $|f(x)-2|≤x^2$. Find the limits: $\lim_{x\to0}f(x)$. $\lim_{x\to0}\frac{f(x)-\sqrt{x+4}}{x}$. I can solve question (a) very ...
2
votes
1answer
32 views

Limit of Series with differences of Floor function

Problem: Evaluate $$ L=\displaystyle \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \left( \lfloor \frac{2n}{k} \rfloor -2\lfloor \frac{n}{k} \rfloor \right)$$. Please help me with this one. I ...
0
votes
2answers
31 views

Product of trigonometric polynomials is a trigonometric polynomial

A trigonometric polynomial was defined as $$f(x) = \frac{a_0}{2} + \sum_{k=1}^{n}(a_k cos(kx) + b_k sin(kx))$$ I heard somewhere that trigonometric polynomials have a ring structure, i.e. a product ...
0
votes
0answers
24 views

Normed space, inner product space, which is larger?

My textbook says normed space is inner product space unless it satisfies some parallelogram identity. In this case, we can conclude inner product space is a subset of normed space. However, norm is ...
0
votes
1answer
26 views

A property of an open ball

Let $X = \mathbb{R}$ and $B(x, \varepsilon)=\{y: |x-y|< \varepsilon\}$. Is it true that for every $x \in X$ and $z\in B(x, \varepsilon)$, we can find a neighbourhood $V$ of $x$ such that if $y \in ...
1
vote
1answer
38 views

Asymptotic approximation of $\int_1^x(1+t^{-1})^tdt$ for $x>1$

I'm self-studying Bruijn (1961)'s Asymptotic Methods in Analysis. Below is the first exercise of Chapter 1. Show that $$ \int_1^x(1+t^{-1})^tdt=ex-\frac{1}{2}e\log x+O(1)\quad(x>1). $$ The ...
1
vote
0answers
14 views

Examples of geometric function without closed analytic form [on hold]

There are some non-closed form algebraic expressions that can be easily expressed via geometric method of form construction. [While what is geometric and what is analytic closed form not intuitive in ...
1
vote
1answer
30 views

Proving power series is uniformly continuous on given interval

Let $$f(x)=\sum_{n=1}^\infty \frac{\sin (nx)}{1+n^3}$$ Prove that $f$ is continuous on $(-\infty,\infty)$. My solution: We first note that $f$ converges uniformly on $(-\infty,\infty)$ since ...
2
votes
1answer
31 views

Limit of $f^{-1}(f(x))$ as $f(x)$ diverges

Let $f(x)$ be a continuous, real-valued, one-to-one function on some domain $(a,b)$ which diverges at one end of the domain (say $b$). As $f$ is one-to-one I expect to be able to define a inverse ...
-3
votes
1answer
64 views

$\mathbb N\times\mathbb N$ is countable

$$\mathbb N\times\mathbb N \text{ is countable}.$$ Is there any way to prove it using induction? without fundamental theorem of arithmetic
1
vote
1answer
67 views

Can any real be found in an interval of $\left(\frac 1 {n+1}, \frac 1 n\right)$?

I am trying to prove that $$\bigcup_{n=1}^\infty\left(\frac 1 {n+1}, \frac 1 n\right)$$ covers $(0, 1)$. To do that, I want to prove that for every real $x$ there exists an interval $\left(\frac 1 ...
4
votes
1answer
38 views

What is a Dynkin system? ($\lambda$-system)

Until recently, all my knowledge of measure theory and Lebesgue integration are from Rudin's book, which focuses solely on the Lebesgue measure, its construction and nothing else. I have just put my ...
2
votes
3answers
99 views

Is there a bijection between $(0,1)$ and $(0,infinity)$ [duplicate]

I tried $\tan(x)$, and $\log(x)$, but seems it does not work, so I wonder is there a bijection or not?
2
votes
0answers
58 views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
0
votes
1answer
53 views

Show that $\{1/n:n∈N\}∪\{0\}$ is compact

The set is in $R^1$ and consists of $0$ and the numbers $1/n$. Call it $E$. Take a set of $n$ intervals of radius $r$, centered less than $2r$ apart and such that $\sum_{i=1}^n r \ge 1/2$. Call the ...
0
votes
1answer
37 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$ a.s.? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
-1
votes
2answers
29 views

Show that there exist $x_{0}\in[0,1]$ such that $f(x_{0})=g(x_{0})$ [on hold]

Let $f,g:[0,1]\rightarrow[0,\infty)$ be continuous such that $\smash{\displaystyle\max_{x \in [0,1]}} f(x) = \smash{\displaystyle\max_{x \in [0,1]}} g(x)$. Show that there exist $x_{0}\in[0,1]$ ...
0
votes
2answers
19 views

Diverging Integrals and pointwise convergence?

I am looking for a $f$ and $f_n$ such that   $(f_n) \rightarrow$ $f$ pointwise on $[0,1]$ $f$ and every $f_n$ integrable but the sequence of integrals $\int_0^1 f_n$ does not converge   ...
1
vote
1answer
23 views

Question about assumptions in proof of continuity of $x^2$

When proving that $x^2$ is continuous we usually show that $\lim_{x \to\ a} x^2 = a^2$ for any x. So we show that for any $\epsilon > 0$, there is a $\delta > 0$ such that $0< | x-a | ...
0
votes
1answer
21 views

Riemann Upper and Lower Sums

Suppose $f$ is defined on $I = [0,10]$ as follows: $f(x) = N$ if $N-1 \leq x < N$, $N$ is an integer. Let $P = \{0,1,2,3,4,5,6,7,8,9,10\} \in \prod(P)$ Find $S[f,P]$ and $s[f,P]$ If $P^* = ...
0
votes
1answer
19 views

Prove this series converges to a continuous function

My problem: Prove that the series $\sum\limits_{n=0}^\infty e^{n(\sin(nx)-2)}$ converges for all $x\in\mathbb{R}$ to a continuous function. By the root test it converges, but as far as the continuous ...
0
votes
2answers
35 views

Two decreasing, convex functions agreeing on a closed set

Fix a closed subset $B$ of $[0,\infty)$ and assume that $0\in B$. I am striving to construct two functions $f,g:[0,\infty)\to\mathbb R$ such that $f(0)=g(0)=1$; $f(x)\geq 0$ and $g(x)\geq0$ for each ...
2
votes
1answer
40 views

Prove that if $f$ and $g$ are integrable on $[a,b]$, then so are $\max{(f,g)}$ and $\min{(f,g)}$

Prove that if $f$ and $g$ are integrable on $[a,b]$, then so are $\max{(f,g)}$ and $\min{(f,g)}$. Since $f$ and $g$ are integrable, we know that $U(f,\mathcal{P})-L(f,\mathcal{P}) < \epsilon$ ...
2
votes
1answer
28 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
0
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0answers
20 views
0
votes
1answer
39 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
0
votes
0answers
17 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
0
votes
0answers
18 views

Clarifications on proof of Doob's Forward Convergence Theorem, warning related to it and proof of a corollary

From Williams' Probability with Martingales: $X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$ --> Is this supposed to be stronger than $\lim X_n$ does not exist because it's ...
2
votes
0answers
25 views

Strong Solutions to Nonlinear ODE by Contraction Mapping

Consider the $1$-d ODE $$-u_{xx}+u-\epsilon u^{2}=f, \tag{1}$$ where $f$ is a nice RHS, say $f\in\mathcal{S}(\mathbb{R})$, and $\epsilon>0$. By using the Bessel potential, one looks for solutions ...
1
vote
0answers
11 views

Space of Riesz transforms is closed

Let $B=\bigoplus_{j=0}^nL^1(\mathbb R^n)$ a Banach space with norm $\|(f_0,\ldots,f_n)\|=\|f_0\|_{L^1}+\cdots+\|f_n\|_{L^1}$. Define $$S=\{(f_0,f_1,\ldots,f_n):f_j=R_jf_0,\quad j=1,2,\ldots,n\}\subset ...
2
votes
2answers
47 views

The Cantor staircase function and related things

The Cantor staircase function https://en.wikipedia.org/wiki/Cantor_function has an interesting property: $\{x\colon f'(x)\neq 0\}$ is a nowheredense nullset. But it it differentiable almost ...
0
votes
3answers
49 views

Finding a Taylor Series Representation: $f(x)= \frac{x}{(1+4x^2)^2}$ [on hold]

Find Taylor series representations for the following function. For precisely what values of $x$ is the series representation valid? $$f(x) = \frac{x}{(1+4x^2)^2}$$
2
votes
1answer
40 views

The set where a derivative vanishes is G-delta

If $f:I\to R$ ($I$ - interval) is differentiable, then $\{x\colon f'(x)=0\}$ is a $G_{\delta}$ set. The lecturer didn't prove this fact and I found no proof in my books. How it can be proven?
-1
votes
0answers
27 views

find the gradient

Let $h:\mathbb{R}^n \rightarrow \mathbb{R}^n$ defined by $h(x) = \phi(|x|^2)x$, where $\phi$ is a real valued function that is differentiable. Find a formula for $\nabla h(x)$ (i.e the gradient of ...
-1
votes
0answers
18 views

Limit of Recursive sequences [on hold]

Given $Z_1 > 0$ and $a > 0$ $Z_{n+1} = (a+Z_n)^{1/2}$ To show $Z_n$ is convergent and find its limit. The general approach would be to use PMI(Induction) to show that there exists an upper ...
1
vote
1answer
42 views

Do there exist bump functions with uniformly bounded derivatives? [duplicate]

Let us consider a bump function $\phi: \mathbb{R} \longrightarrow \mathbb{R}$, smooth, with compact support. The most common examples are built from the function $$ \psi(x) = \begin{cases} \exp ( ...
3
votes
1answer
54 views

Why is $R((X))$ defined as follows?

Let $R$ be a commutative ring. Then $R((X))$ is defined as the set of all $\sum_{n\geq N} a_n X_n$ where $N\in\mathbb{Z}$ and is called "The Formal Laurent series". But why? Why don't we consider ...
4
votes
2answers
117 views

Uniform unboundedness of linear operators

Question: Suppose that $(T_k)_{k=1}^{\infty}$ is a sequence of invertible linear operators on $\mathbb{R}^n$. Suppose that $\forall x \in \mathbb{R}^{n}\setminus \{0\}$, we have $$\lim_{k\to\infty} ...
7
votes
4answers
318 views

Ratio test for sequences, the other direction

Suppose I have a real sequence $x_n\to 0$. Is it true that: $$ \left|\frac{x_{n+1}}{x_n}\right|\to r<1 $$ for some $r\in\mathbb{R}$? If not, is it true that: $$\exists ...