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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9 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
6 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 $S^1$?
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0answers
25 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 ...
0
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1answer
24 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
43 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
50 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
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1answer
20 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 ...
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2answers
37 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
16 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 ...
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1answer
21 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
23 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
45 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
32 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
votes
3answers
51 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', ...
4
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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
votes
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
28 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
20 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
55 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
19 views

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 ?
2
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1answer
28 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 & \\ ...
0
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2answers
24 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
65 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
44 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
10 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
31 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
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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 $[0,\infty)$ ...
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0answers
14 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
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4answers
70 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 ...
-3
votes
2answers
82 views

Under what conditions is $|x+y|=|x|+|y|$ true?

What instance that this equation would be true? $|x+y|=|x|+|y|$ Given that $x$, $y$ are elements of real numbers.
2
votes
2answers
29 views

Finding error in reasoning about geometric series sum

A supposedly fast method to find the sum of a geometric series is the following one. Let $$S = \sum_{n = 0}^{+\infty} q^n$$ then $$S = 1 + q\left[\sum_{n = 0}^{+\infty} q^n\right] = 1 + qS.$$ Hence ...
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0answers
21 views

Stochastic process, Fourier transform and $L^2$

Consider a time domain signal received at a sensor $x(t)$ over some time $t$, and we have performed Fourier transform on $x(t)$ to obtain $X(w)$. While performing Fourier Transform to find the ...
4
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1answer
88 views

Proving a convergence relationship between two sequences

Let $a_{n}$ a sequence of real numbers. Let $\sigma_n= \frac{a_1+a_2+...+a_n}{n}$. Suppose that $\lim_{n\to \infty} \sigma_n=A.$ Prove that $$\lim_{n \to \infty}\frac{1}{\log n} ...
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1answer
37 views

Help me solve this problem. [on hold]

Under what condition is it true that $|x-y|+|y-z|=|x-z|$ ?
4
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0answers
48 views

Fredholm integral?

If one exists, find a continuous, bounded function $f: \mathbb{R} \to \mathbb{R}$ which is not identically zero and which satisfies$$0 = \int_0^\infty K(t, s)f(s)\,ds$$for all $t \in \mathbb{R}$, ...
2
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5answers
121 views

Real Analysis: if the integral of the cube of a function exists, does it follow that the integral of the function also exists?

Let $I=[a,b]$. Given that $\int_a^bf^3(x)\;dx$ exists, does it follow that $\int_a^bf(x)\;dx$ exists? Let's let $a$ and $b$ be real numbers.
4
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1answer
121 views

Find all values such that the series converge:

Find all the values of $p\in\mathbb R$ such that the following series converge: $$\sum_{k=2}^\infty (\log k)^{p\log k}$$ I would like hints only. I've tried using the exponential function ...
0
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1answer
40 views

Prove this sequence is bounded

This is not any exercise on itself, but I was reading a proof in which a sequence similar to this appeared: $$a_n=n\lambda^{n-1}, \quad|\lambda|<1$$ In essence. Then I came across the assertion, ...
2
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2answers
41 views

Show That $\limsup_{x \to x_0} f(x) \ge \liminf_{x \to x_0} f(x)$

My question concerns proving an inequality between two extreme limits, namely: $$\limsup_{x \to x_0} f(x) \ge \liminf_{x \to x_0} f(x)$$ Using the following defintions: Let $f: E \to \mathbb{R}$ be a ...
2
votes
1answer
21 views

Proving that $f \big|_{\partial A} = 0$, where $A = [f> 0]$.

I found this exercise: let $M$ be a metric space, and $f: M \to \Bbb R$ be a function, and $A = \{ x \in M \mid f(x) > 0\}$. Prove that if $x \in \partial A$, then $f(x) = 0$. I think that we must ...
0
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0answers
18 views

A question regarding separable continuity and measurability

Suppose $f(x,y)$ is a function mapping from $R^2$ to $R$ and it is continuous in each variable separately (separable continuity), then why $f(\frac{{\left\lfloor {mx} \right\rfloor ...
18
votes
2answers
171 views

Let $f:K\to K$ with $\|f(x)-f(y)\|\geq ||x-y||$ for all $x,y$. Show that equality holds and that $f$ is surjective.

$K$ is a compact subset of $\Bbb R^n$ and $f:K\rightarrow K $ satisfies : $$\|f(x)-f(y)\|\geq \|x-y\|$$ Show that $f$ is bijective, and that : $$\|f(x)-f(y)\| = \|x-y\| $$ It's easy to show that ...
0
votes
2answers
19 views

Deriving convexity from Taylor series expansion

Why is the function $f(x) = \sum^\infty_{k=1} (3x)^{2k}$ convex? What is the condition on the coefficients to deduce that $f$ convex?
0
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1answer
35 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
2
votes
1answer
40 views

literature on advanced calculus [on hold]

I need your opinions on this particular textbook: Advanced Calculus by Robert C. Buck. In my first year in college I finished two semesters of single-variable calculus and now I'm looking for a proper ...