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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Are there any other non-differentiable that came be constructed from summation besides the Weierstrass function?

So, I'd like to see conditions on a function $f(n,t)$ such that $F(t)$ from, $$F(t)=\sum_n f(n,t)$$ Is continuous over a non-zero range, but is nowhere differentiable. The range of the summation ...
2
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1answer
29 views

The limit of a sequence with two-dimensional indexes?

Given a sequence $x_{m,n}$ with two dimensional indexes $(m,n)∈\Bbb{N}×\Bbb{N}$, what is definition of the limit $\lim_{m,n→∞}x_{m,n}$? The following is my guess, is it right? Thank you! ...
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1answer
30 views

Derivatives Must Exist over an Entire Open Set?

Let the domain of some function $f$ be an open set that contains the point $a$. This question is with regards to the domain of the (partial) derivatives of $f$. Claim: If all the partial derivatives ...
3
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2answers
18 views

Are the stationary points of a strongly convex function unique in each dimension?

Consider a strongly convex function $~f: \mathbb{R}^n \rightarrow \mathbb{R^+}~$ with a unique minimum at the point $x^* \in \mathbb{R}^n$. I am wondering: if I have another point $y \in ...
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0answers
23 views

Finding a lower bound

Given four positive integers $n,$ $m,$ $l$ and $k \geq 2.$ I want to find a lower bound for this expression $$|\sqrt[k]{n}+\sqrt[k]{m}-\sqrt[k]{l}|$$ in terms of these integers. Many thanks
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1answer
41 views

A consequence of Cesàro's theorem

Here is the statement : "Let $(a_n)_{n\ge 1}$ a real or complex sequence and $l \in \bar{\mathbb{R}}$. If $\lim \limits_{n\to +\infty} a_{n+1} - a_{n}=l$, then $\lim \limits_{n\to ...
-1
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1answer
28 views

Supremum vs Integral [on hold]

Let $h$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}h(t)\leq\int_{r}^{\infty}h(t)\frac{dt}{t} $$
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2answers
148 views

Is $e^x$ finite almost everywhere even though $\mathop {\lim }\limits_{x \to \infty } {e^x} = + \infty $?

Does $e^x$ is finite almost everywhere even though $\mathop {\lim }\limits_{x \to \infty } {e^x} = + \infty $? I think it is, but I put on this question to make sure. I know $f$ being finite a.e. ...
2
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1answer
58 views

Easy classical physics made mathematically rigorous!

Consider the following: We are given a symplectic manifold $M$. Now, we define a Hamilton function $H : M \rightarrow \mathbb{R}.$ Additionally, we want that $H^{-1}(x)=:M_x$ is a submanifold. We can ...
3
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0answers
33 views

Proving that $S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$ is open in $C^2[0,1]$ with a specific metric

I am trying to prove that $$S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$$ is open in $C^2[0,1]$ with the metric $d$ given by $$ d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ ...
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2answers
30 views

Showing that a Borel Measure $\mu\equiv 0$

Problem. Let $\mu$ be a Borel measure on $[0,1]$. Assume that $\mu$ and Lebesgue measure $m$ are mutually singular. $\mu([0,t])$ depends continuously on $t$. $f\in L^{1}(\mu)$ for any ...
2
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1answer
35 views

Completeness of ${C^2[0,1]}$ with under a specific metric

Prove that ${C^2[0,1]} $ (set of two times differentiable functions)is complete with metric: $$d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ \sup_{x \in [0,1]}|f'(x)-g'(x)| + \sup_{x \in ...
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1answer
41 views

Is proving $m(E) < \epsilon, \forall \epsilon > 0$ equivalent to prove $m(E) = 0$?

Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a function ...
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2answers
35 views

Parallelogram law in $L_1$ space

Exercise 5.5 from Capinski's and Kopp's book "Measure, Integral and Probability" asks to show that it is impossible to define an inner product on the space $L^1([0,1])$. In order to get this result we ...
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1answer
14 views

A question about regular Borel measures and cumulative distributions.

I am trying to solve the following question but i am almost positive that i need some result that i don't know. I am free to use any measure theoretic tools. Any help is appreciated. If $\lambda$ is ...
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2answers
126 views

IMC 2015 - Problem 10 - Inequality between polynomials and exponential

This is problem 10 from the International Mathematical Competition for University Students of 2015, from day 2, in Bulgaria. I think it is an interesting problem! Let $n$ be a positive integer, and ...
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0answers
27 views

Variable coefficient wave equation

Consider the equation $$u_{tt} - f(x)^{2}u_{xx} + u_{t} = 0$$ for $(x,t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0) = 0$ and $u_{t}(x,0) = 0$ for all $x \in \mathbb{R}$. Furthermore, suppose ...
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1answer
32 views

Notion of fixed point for the sequence $x_{n+1}=f_n(x_n)$

Let $(X,d)$ be a complete metric space and let $f_n:X\to X,\ n\in \mathbb{N}$ be a sequence of contractions. Is there any notion of fixed point for the following sequence? $$x_{n+1}=f_n(x_n),\ ...
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1answer
31 views

Writing line integral as 1-form

If $F: \Bbb R^n \rightarrow \Bbb R^n $ is a vector field and $\phi : [a,b] \rightarrow \Bbb R^n$ is a continously differentiable path we defined the integral of $F$ along $\phi$ as $\int_{\phi} F = ...
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2answers
46 views

How to show that the following series converges to 1

Let $f$ be a function on $\mathbb{R}$, non-zero only on $[0,2)$. In particular $f(x)=1,x\in[0,1]$ and decreasing to zero, starting from $x=1$. Let $g(x)=f(x)-f(2x)$. Show that $$\sum_{j=0}^\infty ...
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1answer
28 views

Intermediary self-learning-readable book(s) for Real Analysis (incl. Measure Theory,…)?

After studying a very readable book, Advanced Calculus by Fitzpatrick, I thought I start more advanced of real analysis by the same author so I started Real Analysis by Fitzpatrick (and Royden). Well, ...
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2answers
37 views

Is it true, that $(a:b) \cdot 2=a:(b:2)$, when b is even?

I was doing my maths homework and I found that $(a:b) \cdot 2=a:(b:2)$ when b is even!I tried to improve it with examples, but I am not sure if it is true.Can you put it to the test and tell me if I ...
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2answers
38 views

How to expand the $\ln(x)$ to Maclaurin series?

There was a silly question - how to expand the $\ln{x}$ to Maclaurin series?
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1answer
42 views

Does this sequence of functions converge uniformly?

So the questions says, let $a_n$ be a sequences of real numbers such that $\limsup |a_n| = 0$. Let $X = [0, 1]$ and for each $n \in \mathbb{N}$ the function $\space$ $f_n :$ $X \mapsto ...
1
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2answers
272 views

Continuity Must Hold in an Entire Open Set?

Claim: If a function $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $\vec a \in \mathbb{R}^n$, it is continuous in some open ball around $\vec a$. Is this claim false? In other words, is it ...
3
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1answer
54 views

Proving that a sequence converges or diverges [on hold]

Prove or disprove that there is a sequence $n_k$ of positive integers (that is not constant) such that both $\cos(2n_k!)$ and $\sin(2n_k!)$ converge. I think that the series diverges but I am not ...
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1answer
33 views

Approximate function as $x$ tends to infinity

I'm looking for a way to approximate the following function $f$ as $x \to \infty$ $$ f = \ln \left( 1 + e^{a_1 x} + e^{a_2 x} + A e^{(a_1+a_2) x} \right) $$ where $a_1$, $a_2$ and $A$ are constants. ...
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1answer
18 views

Proving the last part of Nested interval property implying Axiom of completeness

I took a non-empty set A that is bounded above. And I went on with the regular algorithm, which either gave us a LUB or gave us an infinite chain of nested intervals $I_1$ $\supseteq$ $I_2$ ...
0
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1answer
28 views

if the integrals of a non-negative sequence of functions go to zero, does this imply functions go to zero a.e.? [duplicate]

This question arised when I was dealing with an old qual problem, and if this is true, I'll be done, but I'm not sure if it's true or not: Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of ...
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23 views

Necessary to assume $f\in C^\infty$ in this Fourier transform problem?

Consider the following problem. Is the hypothesis that $f\in C^\infty$ necessary, or could we weaken it and assume just that $f$ is continuous? Let $\hat f$ denote the Fourier transform of the ...
2
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1answer
64 views

Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
8
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1answer
228 views

What star domain has a non-star-domain interior?

Definition: We call a subset $S$ of $\mathbb{R}^n$ a star domain (or star-shaped) if there exists a point $x_0 \in S$ such that for every $x \in S$, the line segment $\overline{x_0x}$ is contained ...
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1answer
37 views

$ |x_{n} - y_{n}| < \frac{1}{n} \Rightarrow |x'_{n} - y'_{n}| < \frac{1}{n}$

This came from a proof on uniform continuity theorem. My textbook claims that if sequence $(x_j)$ and $(y_j)$ on a compact set D have the condition that $ |x_n - y_n| < \frac{1}{n}$, and $(x'_j)$, ...
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1answer
25 views

Dense Domain: Preimage

Given Banach spaces $X$ and $Y$. Regard a bounded operator: $$A\in\mathcal{B}(X,Y)\implies A\in\mathcal{C}(X,Y)$$ Then for dense sets: $$W\leq Y:\quad \overline{W}=Y\implies\overline{A^{-1}W}=X$$ ...
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1answer
282 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
3
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3answers
66 views

Multiplication operator on $L^1$

Let $\phi :X \rightarrow \mathbb{C}$ be measurable with respect to the measure space $(X,\mu)$. Suppose that $\phi f \in L^1(\mu)$ whenever $f \in L^1(\mu)$. Define $M_{\phi}(f)=\phi f$, for $f \in ...
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0answers
28 views

$\liminf_n a_n = \inf_n a_n$ if $a_n \ge a_m$ when $n\mid m$

I would like to ask a reference for the following very easy result: can someone help? Let $(a_n)_{n\ge 1}$ be a sequence of positive reals such that $a_m \le a_n$ whenever $n$ divides $m$. Then $$ ...
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1answer
26 views

Bounded harmonic function on $\mathbb{R}^3$

Any suggestions how to get started? I know Liouville's theorem, but not sure how to apply it here: Let $u$ be a harmonic function on $\mathbb{R}^3$. Assume there exists $C>0$, independent of $x$, ...
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3answers
60 views

Showing that a functions derivative is not bounded on $\mathbb{R}$

Suppose that $f$ is differentiable but not uniformly continuous on $\mathbb{R}$. Prove that $|f'|$ is not bounded on $\mathbb{R}$. So I know that to show that $|f'|$ isn't bounded you would have to ...
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1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
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0answers
27 views

Compute the limit (using LDCT) [duplicate]

Could you please help me with the following? I think it is a Lebesgue Dominated Convergence Theorem but I am not sure. Compute the following: For $1\leq p<\infty$ and $f\in L^p(\mathbb{R})$ ...
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2answers
85 views

Prove that Euclidean distance in $\mathbb{R}^n$ is a distance

I'm trying to show that: $$\forall x,y\in\mathbb{R}^n, d(x,y)=\left(\sum_{i=1}^n(x_i-y_i)^2\right)^{1/2}$$ is a distance. However I have not proved Cauchy-Schwarz yet and I'm pretty sure I wouldn't ...
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5answers
131 views

Show that $(1+p/n)^n$ is a Cauchy sequence for arbitrary $p$

It is a generalization of this question. I am looking for a similar derivation as in here. Can we prove that $(1+p/n)^n$ is a Cauchy sequence for any $p \in [a, b]$ by showing that $$ \Bigg| \left( ...
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0answers
27 views

Show that there exists a $g \in L^1(m)$ s.t. $\phi (F(x))=\int_{0}^{x} g(t)dt$, where $F(x)=\int_{0}^{x} f(t)dt$

Let $m$ be Lebesgue measure on $[0,1]$ and suppose $f \in L^1(m)$ and let $F(x)=\int_{0}^{x} f(t)dt$. Suppose $\phi$ is a Lipschitz function. Show that there exists a $g \in L^1(m)$ s.t. $\phi ...
3
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1answer
27 views

The dual function of composite functions

Given $X$ $Y$ are two finite dimensional Hilbert space. Let $K$: $X\to Y$ be linear and $F$: $Y\to \mathbb R^+$ is convex. Let us use $F^\ast$ to denote the dual (conjugate) function of $F$. Recall $$ ...
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2answers
51 views

Euclidean Spaces: Embedding

Given the real line $\mathbb{R}$ and plane $\mathbb{R}^2$. Are there maps: $$\eta\in\mathcal{C}(\mathbb{R}^2,\mathbb{R}),\vartheta\in\mathcal{C}(\mathbb{R},\mathbb{R}^2):\quad ...
0
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2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
0
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1answer
33 views

Problem in showing that a sequence is a Cauchy sequence on a space with the integral metric.

I'm having difficulty following what is going on and understanding parts in the following example. It is quite similar to an example I posted before (Changing of the limits of integration with the ...
2
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1answer
40 views

Reverse Intermediate Value Theorem

What does it mean to say that a real valued function $ f : [a, b] \rightarrow \mathbb{R} $ is continuous at $ x_0 \in [a, b] $? Assume that $ f : [a, b] \rightarrow \mathbb{R} $ is continuous State, ...
0
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0answers
25 views

The relationship between outer measures and smallest coverings

Recall that if $\mathcal{A}\subset \mathcal{P}(X)$ is an algebra and $\mu_{0}:\mathcal{A} \to [0,\infty]$ is a premeasure on $\mathcal{A}$ then we can define the outer measure $\mu^{*}$ for any set ...