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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Argument verification

Determine all of the accumulation points of the following sets in $R^1$ and decide whether the sets are open or closed or neither. e)All the numbers of the form $2^{-n} + 5^{-m}: (m,n = ...
0
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2answers
25 views

Prove convergence of sequence

A sequence $\{x_n\}$ satisfies $$|x_{n+1}-x_n| \leq \alpha|x_n-x_{n-1} | $$ for $n=2,3,4,\ldots$ and the number $ 0<\alpha<1$ fixed. Prove $\{x_n\}$ is a convergent sequence.
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2answers
18 views

Closure of intersection of sets

Given a set $A$, use $A'$ to denote all its limit points, and $\overline A$ to denote its closure, i.e. $\overline A = A' \cup A$. My question is, given two arbitrary sets $A,B$, does $\overline {A ...
0
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2answers
25 views

Is my procedure correct about sequences?

Let $\alpha\in(0,2)$, and the sequence $$x_{n+1}=\alpha x_n +(1+\alpha)x_{n-1} \quad \forall n\geq 1$$ Find the limit in terms of $\alpha$, $x_0$ and $x_1$. Check my work. If $\alpha=1$, ...
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1answer
25 views

Uniform Convergence of Differentiable Functions

I was going over an old final exam when I came across the following problem. I have a solution (provided by the professor), but it feels really unintuitive. I have thought about this for some time, ...
7
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2answers
74 views

Evaluating the limit of a certain definite integral

Let $\displaystyle f(x)= \lim_{\epsilon \to 0} \frac{1}{\sqrt{\epsilon}}\int_0^x ze^{-(\epsilon)^{-1}\tan^2z}dz$ for $x\in[0,\infty)$. Evaluate $f(x)$ in closed form for all $x\in[0,\infty)$ ...
3
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1answer
38 views

A golden trigonometric diophantine equation

After answering this question I reflected on the identity $$\cos\frac{\pi}{5}=\phi\cos\frac{\pi}{3}$$ and thought of looking for all the quadruplets of positive integers $(a,b,c,d)$ satisfying $$\cos ...
2
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1answer
24 views

A kernel to guarantee integrability

In trying to answer this question, I thought that it might be useful if there exists a function $K:\mathbb R^2\to\mathbb R$ such that for any continuous function $f:\mathbb R\to\mathbb R$, ...
1
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1answer
29 views

Does the formula for arc length hold for other coordinate systems?

Does the formula for arc length, integration of $\sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}$, hold for other coordinate systems, such as cylindrical coordinates, meaning can I compute the integral of ...
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4answers
61 views

When does:$x^{n} ={n!}$ for $x>0$ and $x$ is real number?

How do i solve this equation in $ \mathbb{R}$ :$$x^{n} ={n!}$$, for a real number $x>0 $ ? Note :I have tried to solve that equation in $ \mathbb{R}$ but i didn't succeed ,may should using ...
0
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1answer
63 views

Integers aren't open sets

So I am solving the following question in apostol Determine all accumulation points of the following sets in $R^1$ and decide whether the sets are open or closed (or neither). All integers: The set ...
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3answers
45 views

Prove that if $B = \{x-y : x,y \in A\}$, where $A$ is a Borel measurable subset of $R$ with positive measure

Suppose that $m$ is Lebesgue measure, and $A$ is a Borel measurable subset of $R$ with $m(A) > 0$. Prove that if $B = \{x - y : x,y \in A\}$, then $B$ contains a non-empty open interval centered at ...
2
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2answers
45 views

Please give me an example $d:C[\mathbb{R}]\times‎ C[\mathbb{R}]\longrightarrow\mathbb{R}$ Such that: [on hold]

I need an example $d$ such that: $$d:C[\mathbb{R}]\times‎ C[\mathbb{R}]\longrightarrow\mathbb{R}$$ $$C[\mathbb{R}]=\lbrace f:\mathbb{R}\longrightarrow\mathbb{R}\ | \ f ‎‎\textit{Is differentiable on ...
3
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1answer
18 views

Question on complete spaces, longer, more specific question.

Let $S \subset C^2[0,1]$ (set of two times differentiable functions $f(x)$ on $[0,1]$) which satisfy the following: $$\int_0^1 f(x)\,dx\leq3$$ Question is $(S,d)$ is a complete metric space, ...
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1answer
36 views

show that the function $\{x_n\}\mapsto \sum_{n=1}^\infty 2^{-n}x_n$ is continuous

This problem comes from an old Preliminary exam: Consider the space $[0,1]\times [0,1]\times \cdots$ (the countably infinite product of $[0,1]$ with the product topology) An element of $X$ may be ...
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0answers
44 views

Is this $\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{\cos k\theta}{k} $ alternating series for all values of $\theta$?

I have tried to do other form of alternating series I got this: $$\sum_{k=1,\theta \in \mathbb{R}}^{k=n}\frac{\cos k\theta}{k} $$ Can I say that the above series is alternating series for all ...
0
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1answer
53 views

How to simplify this integrand,

I am trying to compute arc length in three dimensions but am currently stuck with integrating $$\sqrt{1+ e^{-2t} + 4e^{-2t}}$$ Can I get some hints on how to simplify? I didn't combine the second ...
0
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0answers
13 views

Fraction of Lipschitz functions among absolutely continuous ones

Is it true that the space of Lipschitz functions on $S^1$ is a $G_\delta$ subset of the space of absolutely continuous functions on $S^1$? In which topologies ($L^p$, uniform, $C^k$, etc) it is true? ...
0
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0answers
32 views

Trying to prove that if $f:[a, b]\to[s, t]$ is monotone then $f$ is continuous

I'm trying to prove that if $f:[a, b]\to[s, t]$ is monotone (and its image is closed interval) then $f$ is continuous. My attempt: I say wlog, $f$ is increasing. I know that a monotone function only ...
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1answer
32 views

If $d_1(x,y)$ and $d_2(x,y)$ are metrics, prove that $d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$ is a metric.

$$d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$$ The first three properties are trivially proven. The triangle inequality, not so much. I tried using the triangle inequalities that apply to $d_1$ and $d_2$, ...
3
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2answers
51 views

If $\{a_n\}$ converges to $A$, then $\{(a_1\cdots a_n)^{1/n}\}$ converges to $A$

Prove that this sequence converges. I can't do it. Let $\{a_n\}$ be a sequence of positive real numbers that converges to a number $A$. Prove that $\{(a_1\cdots a_n)^{1/n}\}$ converges to $A$.
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1answer
55 views

Is my proof correct about sequences?

Suppose that $\{ a_n\}_n$ is a sequence of real numbers such that $$ (a_{n+1}-a_n) \rightarrow a, \text{ if } \ n \rightarrow \infty. $$ Prove that $$ \frac{a_n}{n} \rightarrow a \, \text{ if } \ ...
3
votes
1answer
23 views

Compact set of real numbers with countably many limit points.

Construct a compact set of real numbers whose limit points form a countable set. My example: Let $E_1=\{1\}\cup \{1+1/n: n\in \mathbb{N}\},$ $E_2=\{1/2\}\cup \{1/2+1/n: n>2\},$ $E_3=\{1/3\}\cup ...
0
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3answers
52 views

Another solution to compute the length of this curve,

What is the length of C, where C is the graph of the function $$f(t) = \frac{e^t + e^{-t}}{2}$$ on the interval $[0,2\pi]$. Is there a nice way to compute this arc length integral, without knowing ...
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0answers
19 views

How does one define the Fourier transform of a probability distribution?

Say $p_X$ and $p_Y$ are two probability distributions on a $m$ element set. Then I see an equality written as, $$\sqrt{m} \vert \vert p_X - p_Y \vert \vert _2 = \sqrt{ \sum_{k=0}^{m-1} \vert ...
0
votes
1answer
23 views

Control of $L^{\infty}$ Norm of 3d Heat Equation Solution for $L^{3}$ Initial Data

Let $w_{t}$ denote the 3-dimensional heat kernel $$w_{t}(x)=(4\pi t)^{-3/2}e^{-\left|y\right|^{2}/(4t)},\qquad y\in\mathbb{R}^{3}, \ t > 0$$ Suppose $f\in L^{3}(\mathbb{R}^{3})$, and let ...
0
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1answer
25 views

Find the interior points of the following set:

I can identify each element of the set $\mathbb{Q}$ $\cap$ $[0,1[$; however, I must confess that it is pretty hard for me to use the proper open ball for it. I would really appreciate your help. Have ...
1
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1answer
48 views

Series convergence proof review (Baby Rudin)

Ch.3 #7. Prove that the convergence of $\sum a_n$ implies the convergence of $$\sum \frac{\sqrt{a_n}}{n},$$ if $a_n \geq 0$. My attempt. If $\sum a_n$ is convergent, then by the root test, ...
0
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1answer
36 views

A closed set is not a submanifold

Can someone explain me why the set $A:=\{(x,y)\in \mathbb R^2: x\geq 0\}$ is not a submanifold? I also got another (easy) question: In our lecture we are always talking about submanifolds. We ...
3
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3answers
53 views

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^3$. Show the equivalence: $f^{(k)}(0) = 0$ for $k=0,1,2 \iff \lim_{x\to 0} \frac{f(x)}{x^3}$ exists

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^3$. Show the equivalence: $$f^{(k)}(0) = 0 \quad k=0,1,2 \iff \lim_{x\to 0} \frac{f(x)}{x^3} \text{, exists.}$$ Trying: Since $f \in C^3$, implies $f, f', ...
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1answer
12 views

Proof that every polilinear map who's domain is $R^{n_1} \times R^{n_2}… \times R^{n_k}$ and co-domain any given real normed space Y is bound.

A Polilinear map\operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
0
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1answer
43 views

Differentiability issue with this function

$f:D\to{R}$ $$f(x)=\frac{1}{x-2}e^{\left|x\right|}$$ Find the domain $D$ of the function and study whether the function is differentiable. Find the left and right derivatives in the points where the ...
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0answers
35 views

{$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$

I have a question that it maybe simple but I can not understand why we have : {$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ ...
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0answers
23 views

Book to learn Darboux integral

What are some good references to , good book to learn Darboux integral ( https://en.wikipedia.org/wiki/Darboux_integral ) ? Please help .
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1answer
56 views

Why is this statement true? [on hold]

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be the function $f(x,y)=(y-x^2)(y-2x^2).$ Why is this statement true: $t\mapsto f(t\xi)$ has in $t=0$ a local minimum for every $\xi\in\mathbb{R}^n$
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1answer
27 views

NOT bounded functions that satisfy a condition.

I am looking for not bounded functions that satisfy a condition. Let $dx$ be a Lebsegue measure on $\mathbb{R}$. Define \begin{align} \mu(A):=\int_{A}\frac{1}{\sqrt ...
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2answers
59 views

Solving this inequality with integral

We have function $f:\mathbb{R}-\{2 \}\to\mathbb{R}$ $$f(x)=\frac{x^2}{x-2}$$ Show that $8\le\int\limits _3^4f\left(x\right)dx\le9$ I solved the definite integral and got $\int\limits ...
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0answers
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Series expansion of reciprocal function of Generalized Exponential Integral

Generalized Exponential Integral of order p has a series expansion http://dlmf.nist.gov/8.19.10 Is there a series expansion of the reciprocal function ?
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1answer
32 views

Example comparing Riemann's and Lebesgue's methods of integration

It is well known that a function which is Riemann integrable is also Lebesgue integrable, and both integrations result in the same value. Question: Can one give an example of a Riemann integrable ...
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0answers
13 views

Type of critical value

Let $f(x,y,z) = (x-y)^2 + e^{z^2}$. Is it correct that the origin is a critical value of $f$ that is a saddle point? I get for the Hessian matrix $\begin {pmatrix} 2 & & \\ & 2 & \\ ...
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2answers
25 views

Radius of convergence of series $\sum^\infty_{n=0} 3^{-n} (2 \pi)^{-n} (\arctan n)^n x^n$

Is it correct that the convergence radius of the series $\sum^\infty_{n=0} 3^{-n} (2 \pi)^{-n} (\arctan n)^n x^n$ equals $12$?
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1answer
50 views

Determine whether the series converges or diverges?

For the series $$\sum_{n=3}^{\infty}\dfrac{3n^2+8n}{7n^3-4n^2+11},$$ I was thinking of using the limit comparison test to $\dfrac{1}{n}$ but is there a better way?
3
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1answer
68 views

Let $A ,B \subseteq \Bbb{R}^{k}$ and $A+B =\{a+b \mid a\in A, b\in B\}$then:

Let $A ,B \subseteq \Bbb{R}^k$ and $A+B =\{a+b \mid a\in A, b\in B\}$then: a)If $A,B$ be open then $A+B =\{a+b \mid a\in A, b\in B\}$ is open. b)If $A,B$ be connect then $A+B$ is connect? c)If ...
0
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1answer
47 views

Problem about integration

Let $\mathcal R$ be a $\sigma$-algebra in a nonempty set $X$, let $\mu$ be a positive measure on $\mathcal R$, let $f:X\to \mathbb C$ be measurable relative to $\mathcal R$,and $f\in L^1(\mu)$. Let ...
0
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1answer
11 views

Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
2
votes
1answer
60 views

How do I show $(a, b)$ is incomplete?

If I take the sequence $\{a + 1/n\}$, then it's Cauchy and the limit as $n$ goes to infinity is $a$, which completes the proof. Is this correct? Edit: I doubt this is the right approach because the ...
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3answers
32 views

Open intervals in $R^1$ is open

I know this question would seem like a duplicate, but I here I provided a proof of the statement I just don't know how to justify certain thing in my proof. Proof: Suppose y is an arbitrarily ...
1
vote
1answer
30 views

Uniform convergency of a two sequence of functions

For, $n\ge 1$ let, $g_n(x)=\sin^2(x+1/n)$ , $x\in [0,\infty)$ and $f_n(x)=\int_0^xg_n(t)\,dt.$ Then, which is(/are) correct ? (A) $\{f_n(x)\} $ converges pointwise to a function $f$ on $[0,\infty)$ ...
0
votes
0answers
15 views

Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
2
votes
4answers
72 views

$0,1,0,1,0,1$… has only $2$ limit points

Prove that the sequence $0,1,0,1,0,1$... has only $2$ limit points : $0$ and $1$. To be frank, I know the solution to the above particular problem. What I am interested is in knowing a general ...