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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12 views

Change of variables from unit unit ball to another ball for integration

What is the general formula for changing coordinates for integration from the unit ball to another ball? For example, if I wanted to change from integrating $f(x-r)$ over $B(0,1)$, the open ball about ...
0
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0answers
11 views

Triangle inequality with a twist

Assume $t>0$ and $x,y,z\in [0,t)$ how would one go about showing $$\min \{|x-y|,t-|x-y|\}\leq\min \{|x-z|,t-|x-z|\}+\min \{|z-y|,t-|z-y|\} $$ If the first one materializes from every minimum, then ...
1
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0answers
8 views

Limit of ratio of quadratic forms

Let $Q_A,Q_B : \mathbb R^n \rightarrow \mathbb R$ be quadratic forms. Find a necessary and sufficient condition for $\lim_{\vec x \rightarrow \vec 0} \frac{Q_A(\vec x)}{Q_B(\vec x)}$ to exist in ...
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1answer
13 views

For every normed space the norm map is not Fréchet differentiable at $0$.

Argue that for every normed space $\mathbb{X} \neq \{ 0 \}$ the norm map $\| \ldotp \|_\mathbb{X} : \mathbb{X} \to \mathbb{R}$ is not Fréchet differentiable at $0$. Not really sure where to start ...
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2answers
12 views

Prove that $C^1[0,1]$ is space of continuously differentaible function with $C_1$ norm is separable.

$C^1[0,1]$ is space of continuously differentiable function with $C_1$ norm.Then the space $ (C^1[0, 1],)$ is a separable space. I am thinking of c^1[0,1] is subset of c[0,1], and c[0,1] is separable. ...
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0answers
17 views

Which values of $p$, $f$ is it differentiable at the point $(0,0)$?

Let $p \geq 1$ and $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} (\sin \|x\|)^p \cos \frac{1}{\|x\|}, & \quad \text{if } \|x\| \not= 0 \\ 0, & \quad ...
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2answers
41 views

Show that $f$ is not differentiable at $(0,0)$ - $\frac{x_1^2x_2}{x_1^2+x_2^2}$

Let the function $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} \frac{x_1^2x_2}{x_1^2+x_2^2}, & \quad \text{if } (x_1,x_2) \not= 0 \\ 0, & \quad ...
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0answers
22 views

Give an example to show that $f_n$ fails to converge to $f$ uniformly over $S$ if $S$ is not compact

Given the theorem: Suppose $S \subset \Bbb R^n$ is compact, and $P$ is an equicontinuous sequence of functions ($f_n$) over $S$ converging pointwise to a function $f$ at each $x \in S$, then $f_n$ ...
3
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0answers
27 views

Check the proof of $\Bbb R$ as set of subsequential limits

I want to prove that there is a sequence in $\Bbb R$ that has all of $\Bbb R$ as its set of subsequential limits. Could someone help me check my proof? If it's not correct, could someone give a proof? ...
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1answer
24 views

Stuck on Applying Cauchy Convergence Criterion - Limit Theory

I get stuck on the following problem for a rather long time that I finally decide to ask for help. The problem is as below: Determine whether the sequence converges by applying Cauchy Convergence ...
5
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4answers
100 views

What is $\lim_{(x,y) \to (0,0)} \arctan(xy)/\sqrt{x^2+y^2}$?

The limit is this: $$\lim\limits_{(x,y) \to (0,0)} \frac{\arctan(xy)}{\sqrt{x^2+y^2}}$$ It's not necessary to give a whole solution, I want the path to see how to solve it. I tried both with ...
3
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2answers
178 views

The set of integers is not open or is open

Baby Rudin gives the example of the set of all integers being not open if it is a subset of $\mathbb{R}^2$ (I forgot how to code the symbols on this site) If we consider the set of integers in ...
1
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1answer
25 views

Pointwise or Uniform convergence?

Consider the functions $f_n:[-1,1]\to\mathbb{R}$ defined by $$f_n(x):= \frac{x}{\sqrt{x^2 + \tfrac 1n}}$$ and determine whether the convergence is uniform or pointwise. I can see that this will ...
-1
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1answer
28 views

Is every topological space is measurable?

Actually I am learning about measure theory. But I have confusion between topological space and measurable . Is there any relationship among them or not?
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1answer
50 views

when $\lim_{n\to\infty }\sum_{k=0}^n\binom{n}{k}x^k$ exist?

When does $$\lim_{n\to\infty }\sum_{k=0}^n\binom{n}{k}x^k\ \ ?$$ My first way : Since $$\sum_{k=0}^n\binom{n}{k}x^k=(1+x)^n$$ the limit exist when $x\in ]-2,0]$ et it's limits is $0$ when $x\in ...
0
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1answer
30 views

$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $ converges pointwise on $ R$, but not uniformly on $R$.

Prove Or Disprove, The series summation $$ \sum_{n=1}^\infty \frac{x^4}{(1+x^4)^n} $$ converges pointwise on $ R$, but not uniformly on $R$. I am struggling on this problem in real analysis ...
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0answers
42 views

how to evalute this equality

I want to prove this equality $$ \frac{1}{2\pi}\frac{(x-y)\cdot y}{(x_1-y_1)^2+(x_2-y_2)^2}= \frac{ab}{4\pi}\frac{1}{a^2\sin^2(\alpha+\beta)+b^2\cos^2(\alpha+\beta)}.\tag{1}$$ where ...
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0answers
25 views

Conditions for a function to lie in $L^p(\mathbb{R})$

Let $(X, \mathfrak{M})$ be a measurable space. What are some sufficient and necessary conditions for a function $f : X \to \mathbb{R}$ to lie in $L^p(\mathbb{R})$ for $p \in [1,\infty]$? Is true ...
2
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0answers
29 views

Proving that $\lim\limits_{t\to\infty} e^{At}x_0 + \int\limits_0^\infty e^{A(t-s)}b(s)ds=\vec{0}$

Consider $x'=Ax+b(t)$, a system of differential equations. Given that $A$ has negative real parts in all its eigenvalues, and that $\lim\limits_{t\to\infty} b(t) = \vec{0}$, I need to prove that ...
0
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1answer
18 views

Oscillation of a function at a point

Why do we need to take open neighbourhoods around the point in consideration while defining oscillation of a function at that point? (We're working in R) For ref. Bartle & sherbert(introduction to ...
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1answer
27 views

Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
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2answers
33 views

prove limit of exponential function without concept of logarithm

The question is, prove that if a real number $x>1$, then $\lim_{n\to\infty}x^n = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure ...
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1answer
36 views

Distinct real roots .

Problem : If $|\log(x)| - px = 0$ has three distinct real roots then the range of $p$ will be ? My attempt : I tried to see the problem graphically and made the graph. So I am able to see that ...
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0answers
20 views

Converting all arcs to polygonal arcs in a plane graph

I am trying to understand a proof on the conversion of arcs to polygonal arcs in plane graphs, in the book "Graphs on Surfaces" by Mohar and Thomassen. In the book, an arc joining two points $x,y \in ...
1
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1answer
25 views

Are these two definition of boundedness equivalent?

Definition 1: A set $S \subset M$ is bounded if $\forall x \in M, \exists r > 0,$ such that $S \subset B_r(x) = \{y \in M | d(x,y) < r\}$ Definition 2: A set $S \subset M$ is bounded if ...
0
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3answers
91 views

Show that $a_n = 1 + \frac{1}{2} + \frac{1}{3} +\dotsb+ \frac{1}{n}$ is not a Cauchy sequence [on hold]

Let $$ a_n = 1 + \frac{1}{2} + \frac{1}{3} + \dotsb + \frac{1}{n} \quad (n \in \mathbb{N}). $$ Show that $a_n$ is not a Cauchy sequence even though $$ \lim_{n \to \infty} a_{n+1} - a_n = 0 ...
1
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1answer
39 views

Is this proof correct? Show $\mathbb{Q}$ is dense in $\mathbb{R}$

I like proof by contradictions in showing that $\mathbb{Q}$ is dense in $\mathbb{R}$. But I can't understand this one> https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf ...
2
votes
1answer
21 views

Show that $f$ is differentiable at point $x \not= (0,0)$ - $h(x) = (\sin ||x||)^p \cos \frac{1}{||x||}$

Let $p \geq 1$ and $f: \mathbb{R^2} \to \mathbb{R}$ defined as $$f(x) = \begin{cases} (\sin \|x\|)^p \cos \frac{1}{\|x\|}, & \quad \text{if } \|x\| \not= 0 \\ 0, & \quad ...
0
votes
1answer
25 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
0
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1answer
25 views

Proof verification: Compact set has sup and inf

I was reading this post compact set always contains its supremum and infimum There was an answer reposted as follows: As $K$ is compact, we have that $K$ is bounded. So $\sup K$ and $\inf K$ ...
0
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1answer
22 views

Show that a sequence $\{s_n\}$ converges to $L$ if and only if the sequence $\{-s_n\}$ converges to $-L$.

I understand that both parts of this biconditional must be proven. If I assume that a sequence $\{s_n\}$ converges to $L$,then for every for every $ϵ>0$, there is some integer N where ...
2
votes
1answer
49 views

To show the $\epsilon-\delta$ definition for limits holds.

Question: Check if the following limit exists, if so show that the $\epsilon$ $\delta$ definition for limits holds. $$\lim_{(x,y) \to (1,2)} \frac{(x-1)^2(y-2)^2}{x^2+y^2-2xy-4y+5}$$ My answer: So ...
1
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0answers
37 views

Show that the sequence of functions $f_n(x)=xe^{-nx}$ for $x\in(0,1)$ converges pointwise to $0$

Is it enough to calculate $\lim_{n \rightarrow \infty} xe^{-nx}$ or should we analyze it more carefully?
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1answer
11 views

Is relative entropy with respect to a pmf a continuous function?

Is the relative entropy $D(p || q)$ with a fixed pmf $q$, continuous over $p$, where $p \in \{x \in \mathbb{R}^n: \sum_{i=1}^n x_i = 1 , x_i \geq 0 \}$?
1
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0answers
39 views

How to prove $\int_{-\infty}^{+\infty}|f(x+t) - f(x)|dx \to 0$ as $t\to 0$

Let $f\in L^1(-\infty, +\infty)$ and $g(t) = \int_{-\infty}^{+\infty}|f(x+t) - f(x)|dx$. How to show that $\lim_{t\to0}g(t) = 0$? Originally I thought that we need to use Dominated Convergence ...
2
votes
5answers
55 views

Prove that $f$ has a minimum

Let $f$ be a positive and continuous function in $[0,\infty)$, such that $\lim\limits_{x\to \infty} f(x)=2$. Prove that if $f(0)<2$, $f$ has a minimum in $[0,\infty)$. I am stuck in the ...
0
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1answer
32 views

Prove that $\sum_{n=1}^{\infty} \frac{[nx]}{n^2} $ is discontinuous at $x \in \mathbb Q$

$[x] := x - \lfloor x \rfloor$. I can prove that it is continuous at all irrational points using uniform convergence, but I don't know how to prove discontinuity in this case. I looked at this similar ...
3
votes
4answers
71 views

If $B\subset A$ and $f:A\to B$ is injective prove it's a bijection between $A$ and $B$

I want to show that if $B\subset A$ and $f:A\to B$ is an injective function then there's a bijection between $A$ and $B$. I believe my "proof" is wrong, I probably use too much "intuition" when I ...
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0answers
45 views

Real Analysis-Find the limit of this series [on hold]

Problem # 3 enter image description here Find the limit of this series.
2
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2answers
44 views

Question about proof: Uniform cauchy $\Rightarrow$ Uniform convergence

I have one quick question regarding the proof of a theorem contained in here : https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch5.pdf Theorem 5.13. A sequence $(f_n)$ of functions $f_n ...
0
votes
1answer
18 views

Growth rate of large sets

Suppose that ${a_k}$ is a real valued increasing sequence such that $$ \sum_{k=1}^{\infty} \frac{1}{a_k} = +\infty ,$$ i.e. $\{a_1,a_2,\ldots\}$ is a large set. If $\lim a_{k+1} - a_{k} = \infty$, ...
2
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2answers
50 views

Proof of $\lim_{h\to 0}\frac{f(a+ h)-f(a)}{h}=\ell$ when $\lim_{x\to a}f'(x)=\ell$

Suppose $f$ derivable on $\mathbb R$ and that $\lim_{x\to a}f'(x)=\ell$. Show that $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\ell.$$ The proof of my course goes like this: By mean value theorem, there is ...
2
votes
2answers
38 views

Several questions about Riesz–Markov–Kakutani representation theorem

This is a list of questions about Riesz–Markov–Kakutani representation theorem . 1)If $f\in L^1(\mu)$, is it true that $\phi(f)=\int_Xfd\mu$, where $\mu$ is given by the theorem? I am quite sure it ...
3
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5answers
42 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
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0answers
24 views

Can I use the Squeeze theorem with sequences of functions?

For example if I know that $f_n(x)\leq g_n(x) \leq h_n(x)$ for all $x$, then can I say that $ \lim_{n \rightarrow \infty} f_n(x)\leq \lim_{n \rightarrow \infty} g_n(x) \leq \lim_{n \rightarrow \infty} ...
2
votes
1answer
33 views

Maximal interval of solutions existence: $x'(t)=-x(t)+\sin x(t)+t^3$

$x'(t)=-x(t)+ \sin x(t)+t^3$ in $\mathbb{R}$ I consider the function: $$ f(t,x)=-x+\sin x + t^3 $$ $$\frac{\partial f}{\partial x}=\cos x-1$$ I see that: $$\left| \frac{\partial f}{\partial x} ...
1
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0answers
11 views

How can I find a mapping $G(x,y)$ [on hold]

How can I find a mapping $G:X\times X \rightarrow X $ (where $X\subset R$) such that $G(x,y)$ is non-increasing in arguments $x$ and $y$ and $G(x,x)=0$ for all $x\in X.$ Thanks
5
votes
2answers
36 views

Deducing the series expansion of $\arctan(x^2)$ via the series expansion of $\arctan(x)$ at $x=0$

Comparing the series expansion of $\arctan(x^2)$ and $\arctan(x)$ at $x=0$ it looks like one can take the result from $\arctan(x)$ and replace each $x$ with $x^2$ to deduce the series expansion of ...
1
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0answers
12 views

About Convergence of a series [on hold]

Is this series convergent? $$\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}n^2}$$ where $c_{n}^{N}$ is coefficient $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. ...
0
votes
0answers
21 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...