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, integration through the fundamental theorem of calculus.

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Suppose $\{e_n\}_{n=1}^\infty $ is the orthonormal system of functions in $L^2[0,1]$, if $e_n \in C^1[0,1]$, then $e_n'(x)$ can be unbounded.

Suppose $\{e_n\}_{n=1}^\infty $ is the orthonormal system of functions in $L^2[0,1]$, if $e_n \in C^1[0,1]$ for all $n\in \mathbb{N}$, then $\sup_n \max_{x\in [0,1]}|e_n'(x)|=\infty$. I consider the ...
4
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1answer
34 views

Can the proof of fixed point theorems ever be constructive?

Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these ...
2
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1answer
40 views

approximation of $\log(1+z)=z$ as $z\to 0$

This is new to me and I have not done any asymptotic approximation. I don't understand how they get that $\frac{n}{N}$ stays close to $\frac{2}{3}$ as N goes to infinity. Also how do they do get ...
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1answer
15 views

Total variation inequality

In an article I've stumbled upon this inequality: $$ V(f) \leq C(f)s(f) $$ where $f$ is a Lipschitz$(1)$ density with Lipschitz constant $C(f)<\infty$ and support $s(f)$. How can this be derived? ...
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1answer
9 views

measurability of supremum of a class of functions

Let $f:X\times Y \mapsto R$ be a measurable function on product space $X\times Y$, where $X$ and $Y$ both are some metric spaces. Define $g(x) := \sup_{y\in Y} f(x,y)$. [Q.] Is $g$ a measurable ...
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1answer
15 views

If a continuous function $f$ satisfying $D^+ f(x) \geq a>0, \forall x\in \mathbb{R}$, where $ a$ is a const, then $f$ is monotone?

If a continuous function $f$ satisfying $D^+ f(x) \geq a>0,\forall x\in \mathbb{R}$, where $ a$ is a const, then $f$ is monotone? Here $D^+ f(x)$ means the Dini's differential, $D^+ ...
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1answer
32 views

A Set That Is “Precisely” Measure-Dense [duplicate]

This question asks for a set of real numbers that is measure-dense, whose complement is also measure dense. In terms of $[0,1]$, the question asks for an $S$ such that for every open interval $I$ we ...
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0answers
13 views

Lipschitz constant bounded and total boundedness

I seem to cannot find it anywhere but if a class of functions has a lipschitz constant that does not exceed a value name $C$. Does it mean the class is totally bounded? Any link or ref would greatly ...
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1answer
52 views

Show the integral is convergent. [on hold]

(a)Show that $$\int_0^1 |\ln(x)| \, dx$$ is convergent. (b)Show that $$\int_0^1 {{x\ln(x)}\over{1+x^2}} dx$$ is convergent. Thank you.
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3answers
68 views

To prove the sum is convergent [duplicate]

Let$$a_n \ge 0$$ for all $n \in\Bbb N$. Show that if $$\sum_{n=1}^\infty a_n$$ converges, then $$\sum_{n=1}^\infty {\sqrt a_n\over n}$$ converges, too. The hint is to expand $$\left(\sqrt a_n-{1\over ...
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0answers
25 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
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0answers
26 views

Lipschitz functions and rotations [on hold]

Suppose that $f$ is a Lipschitz continuous function. Prove that there exists a function $g$ that is a rotation of $f$ such that $g$ is bounded, or find a counterexample.
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3answers
37 views

Composition of Lipschitz functions [on hold]

Suppose that $f$ and $g$ are Lipschitz continuous functions. Prove that $f(g(x))$ is Lipschitz continuous.
7
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1answer
84 views

$\forall x \,\exists k$ s.t. $f^{(k)}(x)=0$, then $f$ is a polynomial

My friend sent me the following problem: Suppose that $f$ is real analytic on $(a,b)$, and that for all $x$ in $(a,b)$ there exists a non-negative integer $k$ such that $f^{(k)}(x)=0$. Show ...
2
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1answer
30 views

Composition and Limits

Suppose that $f$ is a continuously differentiable function with $\lim_{x \rightarrow \infty} f(x)=k$ and $g$ is a Lipschitz continuous function. Prove that $\lim_{x \rightarrow \infty} ...
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1answer
31 views

Every path $f:[a,b] \rightarrow \mathbb {R}^n$ of class $C^1$ is rectifiable, and the lenght $\mathcal {l}$$(f)= \int_{a}^{b} |f'(t)|dt$

Good evening everyone! I am in trouble to demonstrate the folowing theorem: "Every path $f:[a,b] \rightarrow \mathbb {R}^n$ of class $C^1$ is rectifiable, and the length $\mathcal {l}$$(f)= ...
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1answer
32 views

Distance Between Points and Sets

You are given two sets $A, B \subset \mathbb{R}^n$. Show that $\{ x \mid d(x, A) < d(x, B) \}$ is open.
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4answers
162 views

What is an advatage of defining $\mathbb{C}$ as a set containing $\mathbb{R}$?

It is a theorem that every field with least upper bound property and Archimedean property is isomorphic to each other. So it seems not necessary to define $\mathbb{R}$ exactly and we simply denote ...
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0answers
19 views

Describing an open interval I centered at c, $I \subseteq (a, b)$

Entire question: Let (a,b) be an open interval of Real numbers and let $c \in (a,b).$ Describe an open interval I centered at c such that $I \subseteq (a,b)$ I didn't quite get where I should've ...
4
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2answers
85 views

What is $\zeta(n)$ as $n$ tends to $\infty$? How fast it goes to the limit?

What is $\zeta(n)$ as $n\to\infty$? How fast it goes to the limit?
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1answer
16 views

Showing that a function is bounded in $L^1$ given a bound on its distribution function

Let $f \in L^2((0,T)\times\Omega)$ where $\Omega$ is a compact manifold. Suppose I know that for every $k > 0$, $$\mu(\{|f| > k\}) \leq Mk^{-\frac 12}$$ for some constant $M$ (which is ...
2
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0answers
28 views

Multiple differentiability from Taylor expansion

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a real function, and let $0\leq n\leq+\infty$. We make the following assumption: For every $a \in\mathbb{R}$ and for $k=n$ (resp., in the case $n=+\infty$: ...
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1answer
16 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...
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1answer
35 views

A proof about the limit infimum of a bounded sequence

I tried to find a proof for this statement: If we have a bounded sequence $x_n$, then the limit infimum is defined as $a=\liminf_n x_n$ such that $a$ is the largest of real numbers which have the ...
3
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1answer
55 views

How to show a set is compact in a function space?

I have a question asking if $\{f_n\}$ is a compact in $C_b([0,\infty))$ (bounded continuous) with $||\cdot||_{L^\infty}$. The sequence is $$f_n (t) = \sin\sqrt{t+(2n\pi)^2},$$ I have showed that ...
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0answers
14 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
0
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1answer
27 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...
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0answers
30 views

Existence of periodic orthogonal basis in $L^2([0,1])$ which is not trigonometric?

Let $$ \psi(x) := \sin(\pi x). $$ It is well-known that system $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$. My question is the following: Are there other examples ...
2
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0answers
37 views

Can we find a real $s$ such that $f(s)=w$ and $f'(s)≠0$?

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real ...
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1answer
50 views

Is my proof on showing that the set of monotone functions on $[a,b]$ has cardinality of continum correct?

I was given an exercise problem to show that the cardinality of the set of all monotone functions on $[a,b]$ is $\aleph$. I came out with a proof which I am not sure if it is correct. My proof: Let ...
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2answers
36 views

Fixed points of a certain type of functions with intermediate value property

Let $f: \mathbb R\to \mathbb R$ be a function, having intermediate value property, such that $f(f(x))=x , \forall x \in \mathbb R$, then is it true that either the set of fixed points of $f$ is ...
2
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1answer
31 views

A continuous function that attains neither its minimum nor its maximum at any open interval is monotone

Let $f: \mathbb R\to \mathbb R$ be a continuous function such that $f$ attains neither its minimum nor its maximum at any open interval $I \subseteq \mathbb R$ , then how to prove that $f$ is ...
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0answers
34 views

Prove that the function $g(·)$ is twice continuously differentiable and that $g′′(α) ≥ 0$ for all $α ∈ \mathbb{R}$, i.e.

Let $f$ be a real Lebesgue measurable function on the interval $[0, 1]$ such that $∥f∥∞ < ∞.$ For $α ∈ \mathbb{R}$ define a function $g(α)$ by $g(α) = \log \int_0^1\exp[αf(x)] dx .$ (a) Prove that ...
5
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1answer
60 views

If $f: A\to\mathbb R$ one-to-one but not monotone, there exist $x,y,z\in A$ with $x<y<z$ such that $f(x) < f(y)$ and $f(y) > f(z)$ (wlog)

The following result is part of the folklore, but I'd like to have a standard reference for something that I am writing: If $A \subseteq \mathbb R$ and $f: A \to \mathbb R$ is one-to-one but not ...
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0answers
28 views

Differentiability of polynomials

Trivial question but I am confused with the notation If $p_{n-1}$ is a polynomial of degree $n-1$, is it $\in$ the differentiability class C^n$? Obviously if $p_n$ is a polynomial of degree $n$, ...
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0answers
32 views

“Simple” question (Lebesgue Integration) in a hard exam (proof verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
1
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1answer
38 views

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and some condition.

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and ere exists an increasing function $g:[0,1]\rightarrow \mathbb{R}$ such that $$\left|\int_a^b f(x) dx \right|^2 \leq (g(b)-g(a))(b-a)\quad\quad (*)$$ ...
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1answer
22 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
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1answer
35 views

Do all series have a closed form representation of their partial sum? If not, can we feasibly prove that this is not the case?

The question was motivated by the way in which we approach the convergence and divergence of some series. During my undergraduate analysis course one of the only times in which the partial sum was ...
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1answer
18 views

$\int_{[0,1]^2}g(y_1-y_2) \Bbb{1}_{\{y_1>y_2\}}dy_1dy_2 = \int_{[0,1]}g(m)(1-m)\, dm$

i'm trying to prove the following equality $$\int_{[0,1]^2}g(y_1-y_2) \Bbb{1}_{\{y_1>y_2\}}dy_1dy_2 = \int_{[0,1]}g(m)(1-m)\, dm$$ I tried to do the following: ...
0
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1answer
49 views

The definitions of limit infimum and limit supremum

I have begun reading Rosenthal's "A First Look to Rigorous Probability Theory" and in order to reinforce my calculus background I am studying through the appendix section. Here he defined the limit ...
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2answers
66 views

a question about sequence and series. prove $ \lim_{n \to \infty}( n\ln n)a_{n}=0$? [on hold]

Suppose $a_{n}>0$. $na_{n}$ is monotonic, and it approaches 0 as n approaches infinity. $\sum_{n=1}^{\infty} a_{n}$ is convergent. please prove $$ \lim_{n \to \infty} n\ln(n)\,a_{n}=0$$ I totally ...
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0answers
20 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
0
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0answers
29 views

quasi-convexity of a function

Can someone help me identify whether the following function is quasi-convex? Let $p>1$. For $x=(x_1,\dots,x_n),x_i>0,\sum_ix_i=1$, we define $$f(x) = -\log \sum_i (x_i/\|x\|_p)^{p-1}.$$ Plots ...
1
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1answer
36 views

Does every inner product space has an orthogonal basis?

It is proved that every inner product space has a basis $W$, but I am not sure if every inner product space has an orthogonal basis? It is known that every inner space has a maximal orthogonal set ...
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votes
1answer
38 views

Proving the divergence of a sequence [on hold]

Let $\{a_n\}, \{b_n\}$ be sequences of positive numbers. Set $$c_n = \frac{a_n b_n}{b_{n+1}} - a_{n+1}$$ Suppose that $\sum_{n=1}^\infty \frac{1}{a_n}$ is divergent and $\limsup_n c_n < 0$. Prove ...
0
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0answers
25 views

How do you use R to find the box counting dimension of a two dimensional set of data, or scatter plot?

I'm using the software R to do some analysis on some data sets for a graduate project. R has a package called "fractaldim", in this package is a function for finding the box counting dimension. The ...
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votes
0answers
26 views

Two measures having the same moments [duplicate]

Let $\mu_{1}$ and $\mu_{2}$ be two finite Borel measures supported on $[0, 1]$. Suppose $\int_{\mathbb{R}}x^{k}\, d\mu_{1}(x) = \int_{\mathbb{R}}x^{k}\, d\mu_{2}(x)$ for all $k = 0, 1, 2, \ldots$. ...
2
votes
1answer
51 views

Showing that a set is not infinite in measure

Suppose $f_n \geq 0$ for all $n \geq 1$, $f_n \to f$ a.e. on $[0, \infty)$ and there exists a constant $M>0$ such that $$ \sup\limits_{n} \int_{E} f_n(x)dx \leq M \mu(E)$$ for each measurable ...
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votes
1answer
24 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...