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
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47 views

Antiderivative is continuous

The following comes from Bass' book on Real Analysis: (Here $dy$ is Lebesgue measure) Exercise 7.6 Suppose $f:\mathbb{R}\to\mathbb{R}$ is integrable, $a\in \mathbb{R}$, and we define ...
1
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1answer
83 views

A challenge in Prof.Terence Tao's book “Analysis”: Using axiom of specification to define image of a function

On page 64 (3.4 Images and Inverse Images) of "Analysis I" by Terence Tao, it says: Note that the set $f(S)$ ($f$ is a function) is well-defined thanks to the axiom of replacement (Axiom 3.6). ...
0
votes
2answers
35 views

Showing there is a projection between a normed space and a subspace

Problem: Let $E$ be a normed space. Suppose $A$ is a finite dimensional subspace of $E$. Show that there exists a continuous projection $T: E \to A.$ Proof. I can write $E=A\oplus B$, where $B$'s ...
6
votes
1answer
68 views

A limit with the harmonic series

How can we prove the following (similar) limits? $$\sum_{k=1}^n \frac{1}{k} (\ln 2 - \frac{1}{n+2} - \frac{1}{n+3} - \cdots -\frac{1}{2n + 2}) \to 0. $$ $$\sum_{k=1}^n \frac{1}{k} (\ln 3 - ...
2
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0answers
17 views

Existence of Fourier Transform for Implicit function

Given an "explicit" function $f:\mathbb{R}^n\to\mathbb{R}^n$, (e.g $F(x_1,\dots x_n)=\cos(x_n)+x_1^2e^{x_2}$) under some assumptions one can allegedly develop a Fourier transform given by ...
4
votes
1answer
38 views

Order of Growth of a Sum

Let $n>k$ and $r$ be arbitrary positive integers. Define $q=k/n$. I want to show that $$ \sum_{i=0}^{rk} \binom{rn}{i}q^i(1-q)^{rn-i}(rk-i)=\Theta(\sqrt{r}) $$ as $r\rightarrow \infty$. I've ...
3
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2answers
41 views

Difference between boundary point & limit point.

A limit point is just a accumulation point whose neighbourhood contains infinitely many elements of the sequence. Is there any difference between boundary point & limit point? I've read in ...
1
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1answer
28 views

Why is the identity function from $\Bbb R$ with the Euclidean metric to $\Bbb R$ with the discrete metric not continuous?

Using only the definition of sequential continuity, show an example that $f(x) = x: \Bbb R \to \Bbb R'$ is not continuous, where $\Bbb R'$ has the discrete topology. So the definition of ...
1
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1answer
17 views

Sub Sigma-Algebra and measurability

If a random variable $X$ is measurable with respect to a sub $\sigma$-algebra (let's say $\beta_{1}$), such that $\beta_{1}$ $\subset$ $\beta$ , is $X$ -necessarily- measurable with respect to the ...
2
votes
3answers
54 views

Convergence of $\sum_{n=0}^\infty \frac{\ln(1+2^n)}{n^2+x^{2n}}$

Let $x \in \mathbb{R}$. Define the series: $$\sum_{n=0}^\infty \frac{\ln(1+2^n)}{n^2+x^{2n}}.$$ For what $x$ does it converge? It clearly has positive terms. The ratio and root tests seem ...
1
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0answers
20 views

Polynomials of Continuous Bijective Functions Dense in C([a,b])

This is a question that I cam across while practicing some old comprehensive exams at my school and I was wondering if you could offer some advice towards working through it. The question is as ...
2
votes
0answers
43 views

For which values of $\alpha, \beta, x \in \mathbb{R}, x \geq 0$, does the series $\sum_{n=1}^\infty n^ \alpha x^{n^{\beta}}$ converge?

I have to study, for $\alpha, \beta, x \in \mathbb{R}$, $x \geq 0$, the convergence/divergence/irregularity (i.e., when the limit of the $N$-th partial sum does not exist) of the following series: ...
1
vote
2answers
71 views

For which $x \in \mathbb{R}$ does the series $\sum_{n=1}^\infty \frac{x^n}{n!}$ converge?

One problem of my exercise book asks for which $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \frac{x^n}{n!}.$$ The answer given by the exercise book is $|x|\leq 1$, but ...
2
votes
1answer
66 views

Prove $f(x)>\lambda$

Question: Let $\lambda \in \mathbb{R}$. Prove that if $f: X \rightarrow \mathbb{R}$ is continuous at $a$, and $f(a)>\lambda$, then there exists $\delta >0$ such that $f(x)>\lambda$ ...
0
votes
1answer
51 views

Real Analysis: Show that, if P* is a refinement of P, then ||P ∗ || ≤ ||P||.

Suppose P = {x0, x1, x2, ..., xn} is a partition of the interval [a, b]. Define the mesh of P, denoted ||P|| to be the length of the longest partition interval defined by $$ P : \|P\| = ...
0
votes
1answer
39 views

Does the closed form of $f(t) = \int \frac{e^{2 \pi i \alpha t}}{e^{2 \pi i \beta t} - 1} dt$ exist?

I have been working on finding close forms of various Fourier series. The general approach is: From the series find the (not necessarily homogeneous) ordinary differential equation for which the ...
6
votes
1answer
52 views

For which $x\in \mathbb{R}$ does $\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$ converge?

I have to study for which values of $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$$ I was only able to say that the ...
0
votes
0answers
32 views

If the derivative tends to infinity near a point, does that mean that the derivative does not exist at that point?

If $f: [0, \infty) \to \mathbb{R}$, $f \in (C^0 [0, \infty)) \cap C^1(0, \infty)) $ and $\lim_{x \to 0^+}f'\to \infty$ does this imply that the derivative (on the right) $f'(0)$ does not exist?
3
votes
1answer
50 views

Geometrical Interpertation of Cauchy's Mean Value Theorem

Cauchy MVT: If functions f and g are both continuous on the closed interval [a,b], and differentiable on the open interval (a, b), then there exists some c ∈ (a,b), such that $$\frac{f'(c)}{g'(c)}= ...
5
votes
1answer
77 views

For what values of $x$ does the series $\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$ converge?

I have to study the values of $x$ for which $$\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$$ converges. First we say that we must have $x>0$. Then, I have started by rewriting the series as ...
3
votes
0answers
55 views

How much algebra is necessary to understand Rudin's “Real and Complex Analysis”?

I've been reading up on the finite element method, and the text many people recommend is The Mathematical Theory of Finite Element Methods by Brenner and Scott. As part of the background, the authors ...
0
votes
1answer
9 views

Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ (Riemann integral)

Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ where $L(F)$ is the supremum of the lower sums of the Riemann integral. I tried to by contradicction ...
0
votes
1answer
27 views

Why $N= max(2,\frac {2}{\epsilon})$ for $|a_n -L|<\epsilon $ convergence problem [on hold]

Using the proof development strategy used regarding the proposition (for all $\epsilon \in \mathbb{R}^+$ there exists an $N \in \mathbb{R}^+$ such that $|a_n - L| < \epsilon$ for all $n > N $) ...
1
vote
2answers
23 views

Are as constant but not constant random variables trivial sigma-algebra-measurable? Converse?

Are almost surely constant random variables trivial sigma-algebra-measurable? These links suggest no: http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2004&task=show_msg&msg=1121.0001 ...
0
votes
1answer
23 views

Riemann sum and integral approximation error (Lipschitz function)

Given a function is Lipschitz continuous on [0,1] such that ∀ x , y ∈ [0 , 1] , | f(x) - f(y) | ≤ M | x - y | for M ∈ ℝ how would you prove: I tried to use the fact that we can find a point in ...
7
votes
4answers
40 views

For which values of $\alpha \in \mathbb{R}$, does the series $\sum_{n=1}^\infty n^\alpha(\sqrt{n+1} - 2 \sqrt{n} + \sqrt{n-1})$ converge?

How do I study for which values of $\alpha \in \mathbb{R}$ the following series converges? (I have some troubles because of the form [$\infty - \infty$] that arises when taking the limit.) ...
0
votes
1answer
27 views

Understanding density of irrational numbers and Archemedian property

From Density of irrationals I know this much of the proof of the density of irrational numbers "We know that $y-x>0$. By the Archimedean property, there exists a positive integer $n$ such ...
1
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1answer
19 views

Locality of tensors part of definition?

I am wondering whether linearity with respect to scalar functions $f \in C^{\infty}(M, \mathbb{R})$ is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ...
-1
votes
1answer
33 views

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact?

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again ...
3
votes
2answers
62 views

how can I show this integral diverges?

I want to show $E(T_a)=\infty$ $$E(T_a)=\int_0^{\infty}{{x|a|}\over\sqrt{2\pi}}x^{-3/2}e^{-a^2/x}dx$$ to show this I need to show this integral diverges. I know gamma function that $$\Gamma ...
1
vote
1answer
52 views

Convex function inequality for Euclidean norm: $\|(f(x_1),\cdots,f(x_n))\|_2\leq f(\|x\|_2)$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a positive, convex, continuous function such that $f(0)=0$. (If you wish you can also suppose $f$ to be monotone increasing.) I would like to prove or to ...
2
votes
1answer
33 views

Why is this flux zero?

I have the vector field $\vec{F}=(x^2+y+2+z^2, e^{x^2}+y^2, x+3)$ and $S$ the part of the spherical surface $\{x^2+y^2+(z-a)^2=4a^2\}$ that is above the $x,y$-plane, with orientation outwards. I know ...
2
votes
1answer
34 views

A piecewise $C^1$ curve has Jordan measure zero.

$\newcommand{\Reals}{\mathbb{R}}\gamma:[0,1]\to \Reals^2$ is an injective parametrization of a curve $\Gamma$, which is piecewise $C^1$ and the length of the curve is $L(\Gamma_k)<\infty$. 1.1.: ...
2
votes
1answer
39 views

How to tell if a series diverges or is indeterminate? Study of some cases of $\sum_{n=1}^\infty3^n (1+\frac{1}{n})^{n^2}k^n$

Suppose we have a series dependent on a parameter. For example: $$\sum_{n=1}^\infty3^n (1+\frac{1}{n})^{n^2}k^n.$$ By root test, we know that this series absolutely converges (hence converges) if ...
2
votes
1answer
29 views

differentiable and uniform continuity of f and F

Given $f: \Bbb R \to \Bbb R$. define new function: $F(x) =\frac{f(x)-f(a)}{x-a}$ for $x\neq a$. Prove that $f$ is differentiable at $a$ if and only if $F$ is uniformly continuous in some punctured ...
2
votes
2answers
57 views

If $f:X \to [0,1]$ be an onto continuous map and $\{f^{-1} (y)\}$ is compact then Is $X$ compact?

If $f:X \to [0,1]$ is an onto continuous map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again $X$ is ...
0
votes
0answers
23 views

Sums convergent but not uniformly convergent on [0,1]

Show that both $\sum_{n=1}^{\infty} ({1-x}){x^n}$ and $\sum_{n=1}^{\infty} (-1)^n({1-x}){x^n}$ are convergent on [0,1] but only one converges uniformly. Which one? Why? I was playing around with the ...
1
vote
1answer
37 views

How to prove a function is continuous on a compact set?

I´m struggleing with this problem: I know by theorems that inf(d(a,b)) exists if the real value function d is continuous on the set AxB. But how can I prove that d is continuous?
0
votes
1answer
21 views

Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$

I am trying to find a good upper bound on \begin{align*} f(x)={\rm erf}\left(\frac{x+d}{b}\right)-{\rm erf} \left(\frac{x-d}{b}\right) \end{align*} here $d>0$ I know that $f(x)$ is symmetric ...
0
votes
2answers
39 views

is a convex continuous function absolutely continuous

Does a continuous convex function $\mathbb{R} \to \mathbb{R}$ belong to $W^{1,1}_{loc}$ ? thank you.
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0answers
38 views

How to use Stokes theorem [on hold]

Can anybody help me solve this problem? Stokes theorem to get $\oint \vec{F}d\vec{R}$
-2
votes
1answer
36 views

Is p a limit point of the range of {$P_n$} when it converges to p [on hold]

Is $p$ a limit point of the range of {$P_n$} when it converges to p? https://www.youtube.com/watch?v=VEx3Ys6JAJo T/F question (d) in this video. Professor argued: p is a limit point of {$P_b$} but ...
0
votes
2answers
33 views

Proving Integral Test?

Assume that $f(x) \geq 0$ and that $f$ decreases monotonically on $[1, \infty]$. Prove $\int_{1}^{\infty} f(x)dx$ converges iff $\sum_{n=1}^{\infty} f(n)$ converges. My proof: If $f$ is non-negative ...
2
votes
3answers
100 views

Stokes theorem to get $\oint \vec{F}d\vec{R}$

I have the vector field \begin{equation*} \vec{F}=(ye^x,x^2+e^x,z^2e^z) \end{equation*} and the curve $C$ that us given by \begin{equation*} \vec{r}(t)=(1+\cos t, 1+\sin t, 1-\cos t-\sin ...
0
votes
0answers
20 views

Sequence of functions with compact support.

Let $(f_k)_{k \in \mathbb{N}}$ be a sequenz of functions with compact support from $\mathbb{R}^n$ to $\mathbb{R}$ and let $f$ be any function from $\mathbb{R}^n$ to $\mathbb{R}$. a) Define exactly, ...
0
votes
3answers
52 views

What does “all but” means? Rudin 3.2

Rudin thm 3.2 (a) $P_n$ converges to p iff every nbhd of $p$ contains ALL BUT finitely many of the terms of $P_n$
1
vote
1answer
19 views

Divergence theorem to calculate the flux

I have the vector field $\vec{F}(x,y,z)=(-x,-y,z^2)$ and i want to find the flux through the part of the cone $\{z=\sqrt{x^2+y^2}\}$ between the planes $z=1$ and $z=2$. How do I use the divergence ...
-4
votes
1answer
50 views

Direct proof: if $P_n \rightarrow p$ and $P_n \rightarrow p'$, then $p=p'$ [closed]

Direct proof: if $P_n \rightarrow p$ and $P_n \rightarrow p'$, then $p=p'$
1
vote
0answers
30 views

Topological properties of regular and critical points and values

Let $f\colon M\rightarrow N$ be a smooth map between smooth manifolds. Consider the following two statements, the second one under the assumption The set of regular points of $f$ are open in $M$, ...
-1
votes
0answers
34 views

Proofs involving 3 quantifiers: A(3,3)=6 cases [closed]

I knows how to prove statement involving 1 or 2 quantifiers. So there are 6 combinations of 3 universal quantifiers ("for all" and "there exist") with an extra implication that makes a quantified ...