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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2answers
27 views

Is this function increasing?

Suppose $x$ is real and positive. $f''(x) > 0$ (f is convex) is the function $h(x) = (f(x))/x$ increasing? If so, is it strictly increasing? If yes, why? Thank you.
1
vote
1answer
33 views

how to show that $\{x\subset \mathbb R^n: f(x)=b\}$ is closed

(1). let $f: \mathbb R^n \to \mathbb R^m$ be a continuous mapping. Let $b\in \mathbb R^m$. Show $$\{x\subset \mathbb R^n: f(x)=b\}$$ is a closed set. My thought: I want to show that the set ...
0
votes
0answers
14 views

Simply connectedness of spherical shell

Consider a spherical shell $U$ in $R^3$(the open region between two spheres). I want to show that any closed curve in $U$ can be shrunk into a single point without leaving $U$. This exercise appears ...
2
votes
0answers
24 views

How can I show that one of $m(A)$ or $m(\Bbb{R}\setminus A)$ is zero?

Let $A \subseteq \Bbb{R}$ be Borel measurable, and $T$ a dense subset of $\Bbb{R}$. Suppose for every $t \in T$ that $$m((A+t)\setminus A)=0,$$ where $m$ is the Lebesgue measure. Then I want to show ...
2
votes
1answer
39 views

integral involving greatest integer function

Let $S_n = \sum_{k=1}^n \frac{1}{k}$ and $I_n=\int_1^n \frac{x-[x]}{x^2}dx$. Then, what is $S_{10} + T_{10}$? The only clue that i can get is break the limits of integration according as the ...
1
vote
0answers
12 views

Galerkin formulation of 1D linear Schrodinger equation

I am interested in the weak (Galerkin) formulation of the 1D Schrodinger equation: $i u_t−\beta u_{xx}=0$ and $u(x,0)=u_0(x)$. As usual, we do integration by parts which yields: $i (u_t,v)+\beta ...
0
votes
0answers
25 views

$f(x) = 2x \mod 1$ not equal to zero for all $x$?

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?
3
votes
2answers
37 views

Show that the integer nearest to $\frac{n!}{e}$ $(n \geq 2)$ is divisible by $n − 1$ but not by $n$.

Show that the integer nearest to $\frac{n!}{e}$ $(n \geq 2)$ is divisible by $n − 1$ but not by $n$. I am still trying to improve my basic math skills but on this one i did not get far. Taylor ...
1
vote
1answer
13 views

If two functions differ on a set of positive measure, must their essential infima differ, too?

Suppose $f,g : [0,1]^2 \to [0,1]$ are measurable functions differing on a set $P$ of positive Lebesgue measure. Claim: there exists $A, B \subseteq [0,1]$, each of positive measure, such that ...
0
votes
0answers
13 views

Show that $s(x)=\pi(x)/x+\int_1^x \frac{\pi(t)}{t^2}\,dt$

For $x\in\mathbb{R}$, let $\pi(x)=\#\{$ primes $p:p\le x\}$ and let $s(x)=\sum\limits_{\text{primes}} \frac{1}{p}$. Given that: If $a_1, a_2, \dots \in \mathbb{R}$ and $f$ is a $C^1$ function in an ...
1
vote
0answers
11 views

Question about a problem on characteristic polynomial and elliptic in real analysis from Stein and Shakarchi's Real Analysis

I am having some trouble solving a problem in Stein and Shakarchi's Real Analysis Chapter 5. Consider the linear partial differential operator $$L = \sum_{|\alpha|\leq n} a_\alpha ...
0
votes
1answer
16 views

For what $x,y$ does $\sum_{k,l\ge 0} \frac{(k+l)!}{k!l!} \left| x^ky^l\right|$ converge?

For what $x,y$ does $\sum_{k,l\ge 0} \frac{(k+l)!}{k!l!} \left| x^ky^l\right|$ converge? I think that $\sum_{k,l\ge 0}\left| x^ky^l\right|$ will converge for $|x|<1$ and $|y|<1$ since ...
0
votes
2answers
26 views

Show that $\langle\cdot,\cdot\rangle : E \times E \to \mathbb{R}$ is a continuous function

Let $E$ a normed vector space, where the norm is induced by a dot product. The norm of $E \times E$ is defined as $||(x,y)|| = \max\{||x||,||y||\}$. Show that $\langle\cdot,\cdot\rangle : E ...
2
votes
3answers
36 views

how to calculate derivative of $f_n(x)=f \circ f … \circ f(x)$? Derivative on $f \circ f_{n-1}$ or $f_{n-1} \circ f$?

Denote $f_n(x)=f \circ f ... \circ f(x)$, the $n$th power of composition multiplication of $f(x)$. Assume $f(x)$ is differentiable for any order. $f(1)=1$, $f^{'}(1)=p$, $f^{''}(1)=q$ Question: Get ...
0
votes
1answer
15 views

Show that $ ix+\frac{2ix}{e^{2ix}-1} = 1+\sum_{j=1}^\infty \frac{(-1)^jB_{2j}}{2j!}(2x)^{2j}$

I proved that $x\cot x = ix+2ix/(e^{2ix}-1)$, now I need to show that $ ix+\frac{2ix}{e^{2ix}-1} =x\cot x = 1+\sum_{j=1}^\infty \frac{(-1)^jB_{2j}}{2j!}(2x)^{2j}$. I know that ...
1
vote
2answers
15 views

Show that $x\cot x = ix+2ix/(e^{2ix}-1)$

Show that $x\cot x = ix+2ix/(e^{2ix}-1)$ So $x\cot x = x\left( \frac{e^{ix}+e^{-ix}}{2}\cdot \frac{2i}{e^{ix}-e^{-ix}} \right) = \frac{ix(e^{ix}+e^{-ix})}{e^{ix}-e^{-ix}}=\dots$ ...
1
vote
1answer
51 views

Showing a function is differentiable at $x$.

Let $v: \mathbb{R^n} \rightarrow \mathbb{R^m}$ be a function such that $v(y) \neq 0,\forall y \in \mathbb{R^n}$ that is differentiable at $ x \in \mathbb{R^n}$. Show that ...
1
vote
1answer
23 views

convolution is well-defined and differentiable for continuous $f$ and differentiable $g$ with compact support

Let $f$ be a continuous function from $\mathbb{R} \rightarrow \mathbb{R}$. Let $g \in C^1(\mathbb{R})$ with compact support. Prove that the convolution function $$(f*g)(x)=\int f(x+t)g(t) dt$$ is ...
0
votes
1answer
27 views

$S_n = \sum_{k=1}^{n} (\sqrt{1 + \frac{k}{n^2}} - 1)$ Show that $\lim_{n \rightarrow \infty}S_n = \frac{1}{4}$

Show that $\lim_{n \rightarrow \infty}S_n = \frac{1}{4}$ $$S_n = \sum_{k=1}^{n}\left(\sqrt{1 + \frac{k}{n^2}} - 1\right)$$ $$\sum_{k=1}^{n}\left(\sqrt{1 + \frac{k}{n^2}} - 1\right) < ...
0
votes
1answer
30 views

conditions for continuous function

A function $f\colon [0,1]\to [0,\infty)$ is continuous and satisfies $f(0) = \lim_{x\to 0^+}\frac{f(x)}{x}$ und $ f(x)\le\int_0^x \frac{f(s)}{s}ds$ for all $x\in[0,1]$. I'm curious if it implies ...
1
vote
0answers
12 views

Definitions of Continuity and Discreteness of Sets

In what sense (strict definitions/synonyms) is $\mathbb {R} $ continuous? Is this continuity antonymic to discreteness? What are the definitions of these properties?
0
votes
1answer
15 views

Proving a variation of DCT

As homework, I was given the following problem. Suppose $f_n\overset{\text{a.e}}{\rightarrow}f$, and for each $n$ there's a $g_n\in L^1$ satisfying $|f_k|\leq g_k$. Prove that if $g=\lim _n g_n$ is ...
1
vote
2answers
51 views

Verification of this proof that the set of real numbers is uncountable.

I don't want a proof. I just want verification or correction of the proof I supplied: We start with the fact that between any two real numbers there is a rational number. There is an infinite ...
2
votes
1answer
21 views

Are the limits of a.e. equal sequences of measurable functions equal a.e.?

I haven't seen the following fact in any textbook or reference, which either means that it is trivial, or that it's false. Hopefully it is the former. I've attempted a proof: Claim: Let $f_n, g_n : ...
0
votes
1answer
28 views

Proving $f_n\rightarrow f$ such that $\sup_n \| f_n \|_1 \leq K$ implies $\| f \|_1\leq K$

Looking back at my notes from class, I see: Claim. $f_n\rightarrow f$ such that $\sup_n \| f_n \|_1 \leq K$ implies $\| f \|_1\leq K$. It appears after the statement and proof of Fatou's lemma but I ...
-2
votes
0answers
31 views

Exercise 16 in Chapter 5 of Stein and Shakarchi's Real analysis

I'm having trouble with the following problem in Stein's Real Analysis book: Suppose $n$ is the smallest integer $>\frac{d}{2}$. If $$ f \in L^2\mathbb(R^d)$$ and $$(\frac{\partial}{\partial ...
0
votes
2answers
29 views

Vector norm and relationship with euclidean distance

If $y\in E_n$ (n dimensional euclidean space) show that $||\textbf{y}||\leq|\textbf{y}|\leq \sqrt{n}||\textbf{y}||$ Where $||\textbf{y}||$ is the euclidean length of the vector $\textbf{y}$ and ...
0
votes
1answer
29 views

Continuity of $F(x)=\int_{(-\infty,x]}fd\lambda$

For a homework assignment I was told to prove that given $f\in L^1(\mathbb R)$, the following function is continuous $$F(x)=\int_{(-\infty,x]}fd\lambda.$$ I thought to use DCT and show sequential ...
0
votes
1answer
17 views

Proof of 'Possibility of subtraction' from Apostol

I was reading from the book Apostol calculus, There is this theorem Possibility of Subtraction : Given $a$ and $b$, there is exactly one $x$ such that $a+x=b$. The proof is - Given $a$ and $b$, ...
0
votes
1answer
48 views

Proving $1\over x^2$ is not uniformly continuous

I need to show that $1 \over x^2$ is not uniformly continuous on the interval $(0,2]$ using the definition of uniform continuity. Definition of Uniform Continuity on a set A: Let $A \subseteq \Bbb ...
2
votes
0answers
15 views

A sequence of continuous functions which is pointwise convergent to zero and not uniformly convergent on any interval.

The exercise is to construct a sequence of continuous functions $f_n:\mathbb{R}\rightarrow \mathbb{R}, n\in \mathbb{N}$ , which is pointwise convergent to $f(x)=0 , x\in \mathbb{R}$ and not uniformly ...
0
votes
1answer
28 views

Determine whether a property possessed by every term in a convergent sequence is necessarily inherited by the limit.

I'm having difficulty coming up with actual sequences that have the properties below. I've included my thoughts on the questions below. Assume that $(a_n)\rightarrow a$. If every $a_n$ is an upper ...
0
votes
2answers
33 views

Show that $f(x)= x/(e^x -1)+x/2$ is even.

Show that $f(x)= x/(e^x -1)+x/2$ is even. So an even function is such that $f(-x)=f(x)$. So I need to show that $f(x)= (-x)/(e^{-x} -1)+(-x)/2=x/(e^x -1)+x/2=f(x)$. I also know that ...
0
votes
2answers
36 views

Cauchy sequence question

Let $f$ be a continuous function in $ (0,1)$. Determine of the following is right or wrong: If $ (x_n) $ is a Cauchy sequence such that $0< (x_n) <1$ then $f(x_n) $ is a Cauchy sequence. ...
2
votes
1answer
31 views

Poincare' recurrence theorem in measure theory.

I want to propose a problem, it's a version of Poincare' Recurrence Theorem, it's very similar to another problem proposed in this forum, but a bit different: Another version of the Poincaré ...
-2
votes
2answers
79 views

Prove or disprove that $(a_1+a_2+\ldots+a_n)\leq n\sqrt{a_1^2+\ldots+a_n^2}$, by showing that $RHS-LHS\geq 0$ if possible. [on hold]

Prove or disprove that $$\left|a_1\right|+\left|a_2\right|+\ldots+\left|a_n\right|\leq n\sqrt{a_1^2+\ldots+a_n^2}$$ Where $a_1,\ldots,a_n\in\mathbb{R}$ and $n\in\mathbb{N}$. EDIT: I was hoping there ...
2
votes
1answer
19 views

Non-Borel a.e limit of Borel functions

As a homework assignment I'm supposed to prove or disprove Borel measurability is closed under a.e convergence. I think this is not true because the Borel $\sigma$-field is not complete. However, I'm ...
4
votes
1answer
49 views

Show that for $\lambda<0$ we have $\inf(\lambda A)=\lambda \sup(A)$

For $A\subset \mathbb{R}$ and $\lambda \in \mathbb{R}$ let's define: $$ \lambda A = \{\lambda a: a\in A\} $$ I have to prove that for $\lambda<0$ and bounded $A$ we have $\inf(\lambda A)=\lambda ...
0
votes
2answers
54 views

A weaker than “zero derivative” condition implies that a function is constant?

Let $f:(a,b)\to R$ be a continuous function such that $\limsup\limits_{n\to \infty}\frac{f(x_n)-f(x_0)}{|x_n-x_0|}\leq 0$ for every $x_0\in(a,b)$ and sequence $x_n$ converging to $x_0$ such that ...
0
votes
1answer
24 views

Is this operator a distribution?

Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$ a distribution (generalized function)? I need to check two things: whether ...
0
votes
2answers
23 views

Fourier transform problem with symmetric matrix. Related to Gaussian?

Hi everyone I encountered a problem that looks simple enough but I have no idea where to start. Find Fourier transform of $e^{-\langle Ax,x\rangle}$ when $A$ is a positive definite symmetric $n ...
1
vote
0answers
21 views

To what Sobolev space does this function belong to?

I am given this function: $$f(x) = e^{- \sqrt{|x|}}$$ and I want to find $k\in \mathbb{N}, \ p \ge 1$ such that $f \in W^{kp} (\mathbb{R})= \{ f \in L^p (\mathbb{R}) \ | \ \forall \alpha \le k: \ ...
0
votes
0answers
15 views

Testing convergence of a sum with two improper integral in it

For wich $\alpha$ does the series converge?$$\sum_{n=3}^{\infty}\sin{\{2\pi n^2 + [\int_{(\ln n)^\alpha}^\infty arctan(t)*(\sin {1/t})^3 dt]*[\int_{\ln n}^\infty \frac{arctan(t^2)}{e^t +2}dt]\}}$$ i ...
0
votes
5answers
50 views

Trigonometric proof/identity (real analysis)

Suppose that $z+\frac{1}{z}=2cos\theta$, where $\theta$ is a real number. Then show that, for any integer $n$, $$z^n+\frac{1}{z^n}=2cos(n\theta)$$ My first thought was to use induction to achieve the ...
2
votes
0answers
20 views

Measurable real functions from $\sigma$-algebra generated by finite partitions

I was given the following homework problem. Let $f:X\rightarrow \bar{\mathbb{R}}$ a set function and $X$ be a measurable space whose $\sigma$-algebra is generated by a finite partition $E_1,\dots ...
32
votes
14answers
3k views

Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that non of the $a_k$'s is a linear ...
2
votes
1answer
41 views

Find a continuous PDF on $[0,6]$ for given probabilities

Find a continuous probability density function $f$ on $[0,6]$, such that $\mathbb{P}([0,2]) = 0.6$, $\mathbb{P}([1,4]) = 0.5$ and $\mathbb{P}([3,5]) = 0.2$. After some calculations I came up with ...
-1
votes
0answers
11 views

Study of heat equation on a torus [on hold]

Good morning, can someone please help me to find a web site where there is the study of heat equation on a torus. thanks in advance
2
votes
2answers
38 views

Numerical evaluation of first and second derivative

We start with the following function $g: (0,\infty)\rightarrow [0,\infty)$, $$ g(x)=x+2x^{-\frac{1}{2}}-3.$$ From this function we need a 'smooth' square-root. Thus, we check $g(1)=0$, ...
0
votes
1answer
15 views

How to show map is non-singular

Let $f:\;\mathbb{R}^n\to\mathbb{R}^n$ be differentiable. Suppose that for all $x\in\mathbb{R}^n:$ $$\lVert \mathrm{D}f(x)-\mathrm{I}\rVert\leq \frac{1}{2}$$ where $\lVert\cdot\rVert$ is the ...