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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0
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1answer
8 views

Be $f:\mathbb{R}^{2}\to \mathbb{R}^{2}$ a continuous function and $g(x)=\int_0^1 \! f(x,y) \, \mathrm{d}y.$ Proves that g is continuous.

I don't see how to solve the following problem, I think that it's like a generalization of the fundamental theorem of calculus. Be $f:\mathbb{R}^{2}\to \mathbb{R}^{2}$ a continuous function and $g(...
2
votes
3answers
52 views

Construct a subset in $\mathbb{R}^{2}$ that is not Lebesgue measurable

Construct directly a subset in $\mathbb{R}^{2}$ that is not Lebesgue measurable. (Don't use the corresponding result in $\mathbb{R}$). Since I could not use the result in $\mathbb{R}$ where we can ...
0
votes
1answer
44 views

Lusin's Theorem: Can we assume nested sets?

This is the statement of Lusin's Theorem (taken from Royden): Let $f$ be a real-valued measurable function on $E$. Then for each $\epsilon>0$, there is a continuous function $g$ on $\mathbb{R}$ ...
4
votes
1answer
52 views

Finding limit of sequence.

$k$ is nonnegative integer. I want to show that$$ \lim _{n \to \infty} {\frac{n!}{n^k(n-k)!}}=1$$ My try : $$ \frac{n!}{n^k(n-k)!} = \frac{n}{n} \frac{n-1}{n} \cdots \frac{n-k+1}{n}$$ I wanted use ...
1
vote
1answer
21 views

sum of two sequences of functions converging in measure still converges in measure

Suppose $f_n\to f$ in measure and $g_n\to g$ in measure. Can I claim that $(f_n+g_n)\to f+g$ in measure? Attempt at the proof: Since we know that $f_n$ and $g_n$ converge in measure respectively, we ...
0
votes
2answers
44 views

How to show that $e^{-\frac{1}{x}}<x^n$ within $0<x<\delta$?

I would just like to show that, given a positive integer $n$, it is possible to find a positive real number $\delta$ such that $$e^{-\frac{1}{x}}<x^n,~~~~0<x<\delta$$ For various values of $...
4
votes
0answers
48 views

are immersions of surface in $\mathbb R^3$ dense in all regular maps?

Let $u\in C^\infty(\Omega,\mathbf R^3)$ with $\Omega$ open set in $\mathbf R^2$. Can we find $u_k\in C^\infty(\Omega,\mathbf R^3)$ with $\mathrm{rank}(Du_k(x))=2$ for all $x\in\Omega$ such that $u_k \...
1
vote
1answer
11 views

Addition of $X^*$ is continuous w.r.t. Weak Topology $\sigma(X^*, X)$

I showed that the addition on $X^*$ is continuous w.r.t. Weak Topology $\sigma(X^*, X)$. Since I'm a newbie in this field, would you please check my proof and answer that my proof is correct or not? ...
1
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2answers
40 views

Please verify my epsilon - delta proof $\lim_{x\to 2}⁡(x^3 )=8,$ and $0 < x < 4$

I am concern about the delta calculation given $0 < x < 4$. I believe it works, but I am not 100 percent sure as I am new to proofs. Consider the function $f(x) = x^3$ for $x ∈ ℝ$ and $0 < x ...
1
vote
1answer
39 views

Partial sum of bounded series is Cauchy in $C(X, \mathbb{R})$?

I am reading a proof of Tietze's Extension Theorem and there was a claim that, given a sequence of functions $h_n(x) : X \to \mathbb{R}$ If $$G = \sum\limits_{n = 1}^\infty h_n(x)$$ Is bounded, then ...
8
votes
2answers
264 views

Show $\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\big( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\beta}}\big)$

Hi I am trying to prove this interesting integral $$ \mathcal{I}:=\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\left( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\...
1
vote
1answer
381 views

A collection of pairwise disjoint open intervals must be countable

Let $U$ be a collection of pairwise disjoint open intervals. That is, members of $U$ are open intervals in $\mathbb{R}$ and any two distinct members of $U$ are disjoint. Show that $U$ is countable. ...
2
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0answers
46 views
+100

Good closed form approximation for iterates of $x^2+(1-x^2)x$

Let $f(x) := x^2+(1-x^2)x$. Is there a nice nontrivial closed form approximation $g_n(x)$ over $[0,1]$ for the $n$-fold composition $f^{\circ n}(x)$? Obviously near $0$ we have that $f^{\circ n}(x) = ...
0
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0answers
8 views

Rigorious formulation of approximation of integral as area of a square and its radius of convergence

We know that the taylor expansion of $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...
3
votes
2answers
28 views

Continuous functions with orbit of period $3$

I would like to build some continuous functions $f : E \to \Bbb R$ (where $E \subset \Bbb R$ is an interval), such that $$\exists x \in E,\;\; [f(x)≠x≠f(f(x)),\;\; f^3(x):= f(f(f(x)))=x]$$ I tried ...
0
votes
1answer
36 views

epsilon delta proof that limit $x+t^3$ as t approaches a is $x+a^3$

Found a small exercise on page 101 of Spivak's Calculus. I'm having some trouble with a clean proof that works even when $a = 0$ that doesn't rely on limit theorems. My naive approach: $$|{f(t)-l}| = |...
0
votes
0answers
26 views

On a certain inequality for the power sum

Let $a_k \ge 0$ and $b_k \ge 0$ for $ k \in N$ such that the following two conditions (i) $0 \le \sum_{k=1}^{\infty}\frac{a_k-b_k}{k^{\alpha}}$ for $0<\alpha \le \alpha_0 <1$, (ii) $ \...
0
votes
2answers
42 views

Function that is onto and continous

Can there be a continuous onto function $f : \mathbb R \to \mathbb R \setminus \mathbb Q$? I know if a function existed it would map reals to the irrationals. Any ideas?
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0answers
19 views

Graph of a subset of $R^n$

I would like to know the definition of a graph of a subset of $\mathbb{R}^n$ which is a set. Each time, I found links about graph theory but here is my context:
0
votes
1answer
21 views

When does a sequence (or a series) of real-analytic functions converge to a real-analytic function?

It is well known that a sequence (or a series) of holomorphic functions converging uniformly converges to a holomorphic function. I would like to know under what condition a sequence (or a series) or ...
3
votes
3answers
124 views

Uniform Continuity on an unbounded domain

The function $f(x) = sin(x)cos(x)$ is continuous on $(-\infty, \infty)$. Is it uniformly continuous? -No, since the domain is not closed or bounded, the domain is not compact, thus the function is ...
0
votes
0answers
39 views

Rigorious formulation of approximation of integral as square for large 2nd derivative.

We know that the taylor expansion of $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...
10
votes
2answers
260 views

Evaluating $ \lim_{n\rightarrow\infty} n \int_{0}^{1} \frac{{x}^{n-2}}{{x}^{2n}+x^n+1} \mbox {d}x$

Evaluating $$L = \lim_{n\rightarrow\infty} n \int_{0}^{1} \frac{{x}^{n-2}}{{x}^{2n}+x^n+1} \mbox {d}x$$
1
vote
2answers
28 views

Showing $f(a) \in V$ but $f(x_n) \notin V$ for every $n$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $f: X \rightarrow Y$ be a function. Let $a \in X$ and suppose $f$ is not continuous in $a$. Prove that there exists an open subset $V$ in $Y$ ...
2
votes
3answers
116 views

How can I show this sequence $u_n$ is divergent: $u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))$

How can I show this sequence $u_n$ is divergent: $$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))\quad n\in \mathbb{N}^*;\quad \epsilon \in (0,1)$$ My attempts: \begin{align*} u_n&=\exp( n\log ...
2
votes
1answer
78 views

Find the value of $\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$

Determine if the following limits exist $$\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$$ note that $$\frac{1}{x}-1 <\lfloor \frac{1}{x}\rfloor \leq \frac{1}{x}$$ $$1-x <x\lfloor ...
5
votes
2answers
98 views

Show that $\lim_{n\to\infty}\frac{\lfloor nx \rfloor}{n}=x$

How do I prove that $\lim_{n\to\infty}\frac{\lfloor nx \rfloor}{n}=x$ for $x\in\mathbb{R}$? I see that $\lfloor nx\rfloor = n\lfloor x \rfloor + \lfloor n(x-\lfloor x \rfloor )\rfloor + O(1)$ but I'm ...
11
votes
3answers
282 views

Solve $\lim_{x\to +\infty}\frac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$

Determine if the following limits exist $$\lim_{x\to +\infty}\dfrac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$$ note that $\lfloor x \rfloor \leq x < \lfloor x \rfloor + 1 \implies x-1 ...
17
votes
3answers
470 views

Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$

Hi I am trying to prove the relation $$ I:=\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}. $$ I tried expanding the log argument by using $\sin x/ \cos x=\tan x,$ and than used $\log(...
5
votes
1answer
250 views

Show that $\int_0^\infty \frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4} dx=\frac{\pi}{2}e^{-a}+\cosh a\log \coth \frac{a}{2}+2\sinh a \arctan(e^{-a})-2$

Hi I am trying to prove this $$ \int_0^\infty \frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4} dx=\frac{\pi}{2}e^{-a}+\cosh a\log \coth \frac{a}{2}+2\sinh a \arctan(e^{-a})-2,\qquad a>0. $$ What a ...
4
votes
2answers
173 views

Coefficient Calculation on Fourier Series Under two minutes, Yes, How?!

Example of one Question for preparing the entrance exam: Fourier series of function: $$ f(x)=f(x+2\pi), f(x) =\left\{ \begin{array}{rcr} 1 & & -\pi <x<0 \\ \sin x & &...
4
votes
6answers
268 views

How to Find a rational number between two irratonal number? [on hold]

Find the rational number between $\sqrt 2$ and $\sqrt3$. I try to solve by using some methods in my book but can not understand steeps.
1
vote
0answers
14 views

Is it possible to use Lusin's Theorem to derive Frechet's Theorem?

Frechet's theorem states that every measurable function $f$ on $\mathbb{R}$ is the limit of a sequence of continuous functions converging almost everywhere. Frechet's theorem is then used to prove ...
-1
votes
0answers
28 views

Creating a “nice” R^2->R function from partial subdomains

This question is probably poorly defined, mainly because I am still not entirely sure what I need, and I just need some ideas to start with. I have a R^2 domain, with a set of given curves ...
22
votes
1answer
239 views

How smooth can non-nice associative operations on the reals be?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is "no" -- a ...
0
votes
1answer
40 views

Density of $C^\infty$ functions on $C[-1,1]$

I'm reading Peter Lax functional analysis, and the author assess that that there exits for every $f \in C[-1,1]$ so that $f(0)= 0$ a $C^\infty$ function $g$ such that the distance between $f$ and $g$ ...
0
votes
1answer
17 views

About uniformly integrability

This problem is from a exercise where I want to apply Vitali theorem.So I'm trying to prove that: If $\left\{f_{n}\right\}\subseteq L^{p}$ ($1\leq p <\infty$) where $\left\{\left\|f_{n}\right\|_{L^{...
1
vote
1answer
28 views

showing $\int f_n^+\to \int f^+$

Supose $(f_n)$ be such that $\int|f_n-f|\to 0$, where $(f_n)$ is Lebesgue integrable. Show that $\int_E f_n \to \int_E f$ for all Lebesgue measurable sets $E$, and furthermore that $\int f_n^+\to \int ...
0
votes
1answer
60 views

Prove $s_1 = 4, s_{n+1} = \sqrt{3s_n -2}$ Convergent Sequence

Prove that this sequence is convergent, as $n -> \infty$ $s_1 = 4, s_{n+1} = \sqrt{3s_n -2}$ In our lecture, our teacher explained proving the above using the following theorem: A ...
0
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0answers
24 views

On Riemann's series of functions

The questions given below are devoted to study properties of some series of functions which are directly connected with Riemann's Conjecture(see, for example Again near at Riemann ...
0
votes
2answers
36 views

Prove a composition of two functions is meaurable [duplicate]

Let $f,g:[0,1]\rightarrow [0,1]$ be measurable functions.Is $g\circ f$ measurable or not? The composition is definitely measurable from the axiom definition of measurable function. But if we want to ...
0
votes
4answers
62 views

Prove: if $s_n$ is Cauchy Sequence, then $s_n$ converges as $n \to \infty$

How to prove the following: if $s_n$ is Cauchy Sequence, then $s_n$ converges as $n \to \infty$ Using the following theorem: A Cauchy Sequence is bounded. Assumption: 1. $s_n$ is ...
1
vote
1answer
36 views

A Question on Normal Operators

Let $T$ be a compact operator on a Hilbert space $H$. I want to prove the following: If there exists Orthonormal Basis from the eigen vectors of $T$, then $T$ is normal operator.
4
votes
1answer
78 views

Proving that $\int_a^bf^pd\alpha=0$ implies $\int_a^bfd\alpha=0$ for $f \in \mathcal{R}(\alpha)$ and $f\ge0$ on $[a,b]$

The question posed may seem trivial to many. But I couldn't find a trivial solution to above within the theory upto Riemann-Stieltjes integral. I woudn't be considering arguments from Lebesgue theory ...
1
vote
0answers
35 views

Taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ Is the taylor expansion of $\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$ at $t$ = $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = ...
2
votes
1answer
23 views

How do I prove two definitions of the variation of a measure are equivalent?

Let $(X,\Sigma)$ be a measurable space and $\mu:\Sigma\rightarrow \mathbb{C}$ be a complex measure. Define $|\mu|(E)$ as the supremum of $\sum_{n=0}^\infty |\mu(E_n)|$ where $\{E_n\}$ is a mutually ...
1
vote
4answers
35 views

Question Involving Open/Closed Sets [on hold]

Let $f : X \rightarrow \mathbb{R}$ be a continuous function and let $a \in \mathbb{R}$. Determine whether the following statements are true and false. Prove you answer. i) $\{x \in X :f(x) \leq a\}$ ...
1
vote
2answers
38 views

Proof of limits involving the epsilon/delta with 0 ≤ x ≤ 8, ≤ confusion

I need help with a proof similar to the following: Consider the function $f(x) = x^2$ for $0 \leq x \leq 8$. Prove that $ \lim_{x \to 4} ⁡f(x)=16$. I understand how to do it for $0 < x < ...
5
votes
1answer
92 views
+50

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
0
votes
1answer
11 views

Non empty interior in the image implies open map

I am looking at the proof showing that $L^2(0,1)$ is meager in $L^1(0,1)$. Define $B_n = \{f\in L^2 : \|f\|_2 \leq n\}$. With the continuous identity map $T:L^2 \rightarrow L^1$, if one of $T(B_n)$ ...