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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3
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1answer
67 views

How prove $\sqrt{x}+\sqrt{y}+\sqrt{\frac{x+y+2}{xy-1}}\ge 2\left( \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}+\sqrt{\frac{xy-1}{x+y+2}}\right)$

How prove $\sqrt{x}+\sqrt{y}+\sqrt{\frac{x+y+2}{xy-1}}\ge 2\left( \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}+\sqrt{\frac{xy-1}{x+y+2}}\right)$ for $8x\ge13, 8y \ge 13$?
1
vote
1answer
49 views

Negation of sequence convergence

I just to verify that the negation of sequence convergence of ($a_n$) to a limit, a would be something like: "There exists a $ \epsilon >0$ s.t. for all $n \in \mathbb{N}$ there exists an $n> ...
0
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0answers
112 views

Proof of the cardinality of continuous functions from $[0,1]$ to $[0,1]$.

I've been thinking about the cardinality of continuous functions from $[0,1]$ to $[0,1]$. I know that the cardinality is the same as that of $[0,1]$ and the standard proof using the fact that such a ...
0
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0answers
55 views

Prove that $p \mapsto \inf\{d(p,s) : s \in S\}$ is uniformly continuous

From Pugh's Real Mathematical Analysis, #14, p. 115: The distance from a point $p$ in a metric space $M$ to a non-empty subset $S \subset M$ is defined to be $\text{dist}(p, S) = \inf\{ d(p,s) : s ...
0
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1answer
28 views

If a function $F$ belongs to mixed Lebesgue space $L^{p,1}$, does its reflection $G(x,y):=F(y,x)$ also lie in the space?

Let $F:\mathbb R^{2}\to \mathbb C$ be a function. Suppose $F\in L^{p,1}(\mathbb R \times \mathbb R); (1<p< \infty).$ Define $G:\mathbb R^{2}\to \mathbb C$ as follows: $$G(x,y):=F(y,x)$$ My ...
1
vote
1answer
28 views

the cardinality $|\{A_{n}\}| = c$ where $c$ is the continuum

Suppose I let $\mathbb{R}^{\mathbb{N}}$ denote the set of all infinite sequences of real numbers. I can't seem to wrap my head around the idea that the cardinality $|\mathbb{R}^{\mathbb{N}}| = c$ ...
1
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1answer
65 views

Is $\{1,\frac{1}{2}, (\frac{1}{2}+\frac{1}{3}), \frac{1}{12}, (\frac{1}{12} + \frac{1}{5}), \dots\}$ dense in $[0,1]$?

This is a problem which emerged from trying to prove that a sequence of random variables on $[0,1]$ does not converge almost surely to $0$. In the process of trying to prove the above, I encountered ...
0
votes
2answers
56 views

Showing that an indicator function of open intervals is Borel measurable

I have the following exercise from the book "A course on real analysis": Show that $\chi_I$ is Borel measurable for each open interval $I$, where $\chi_I$ denotes the indicator function of $I$. ...
0
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0answers
50 views

Hessian Matrix Identity

I'm trying to see why the following identity is true. I've lifted it from an exercise in Miles Reid Algebraic Geometry. I'm having difficult deriving it and feel there might be a simple line or two to ...
4
votes
2answers
101 views

Prove $\exists$ closed subset A st $f(A)=A$

Let $(X,d)$ be a compact metric space and $f$ be continuous on $X$, show that there exists non-empty closed subset $A$ of $X$ such that $f(A)=A$ So, $f$ will be uniform continuous, but how does ...
3
votes
1answer
203 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $2\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
0
votes
2answers
49 views

Is it possible to do this? Write a fraction as a product

I have two quantities $A$ and $B$ and I consider the fraction $$\frac{1}{A+B}$$ I would like to write the above expression as a Product, i.e. find functions $F$ and $G$ such that $$\frac{1}{A+B} = ...
2
votes
1answer
46 views

Prove that $\sum ( \prod_{i=1}^n \frac{2i-1}{2i} )^3$ converges

It is asked to prove that the series $$\sum_n (\prod_{i=1}^n \frac{2i-1}{2i})^3$$ converges. Unfortunately, the ratio test is not conclusive, so I am trying to apply the comparasion test. I've noted ...
5
votes
1answer
120 views

Is $C^{\infty}([0,1])$ a Banach space?

I have read that the answer is no, but I am unable to prove it. Give $C^{\infty}([0,1])$ the metric $$d(f,g) = \sum_{j=0}^{\infty} 2^{-j} \frac{||(f-g)^{(j)}||}{1 + ||(f-g)^{(j)}||}$$ associated to ...
4
votes
5answers
94 views

$\mathbb{C}$ and $\mathbb{R}^{2}$

If a complex number is a pair of real numbers, then why we need to introduce the new symbol $\mathbb{C}$ for complexes instead of using $\mathbb{R}^{2}$? What are the subtle differences involved?
1
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2answers
29 views

Verification of sequence result

Is it true that if a real sequence $\{x_n\}_1^\infty$ has an infimum but no convergent subsequences then the infimum must be the minimum as well? Secondly, can it be proved that the sequence defined ...
3
votes
1answer
78 views

Ad-hoc proof of convergence of $\sum_{n=0}^\infty \sin(\pi\sqrt{n^2+a^2})$

I use the following method to prove that $$\sum_{n=0}^\infty \sin(\pi\sqrt{n^2+a^2})$$ converges for any $a \in \mathbb{R}$. First, we can see that $$\lim_{n\to\infty} \frac{\sqrt{n^2 + a^2}}{n} = ...
1
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1answer
50 views

AN application of Schwarz inequality.

In the proof of Chung-Erd$\ddot{o}$s inequality: Let $X_k=1_{A_k}$,then: $$(\mathbb E(\sum_{i=1}^nX_k))^2\le\mathbb P(\sum_{i=1}^nX_k\gt0)\mathbb E[(\sum_{i=1}^nX_k)^2]$$ The textbook said this ...
0
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0answers
30 views

Error ? A subset $A$ of $ \mathbb R^p$ S.T $A^o = \phi$ and $A^- = \mathbb R^p$ where $A^o$ is interior of $A$ and $A^-$ is closure of $A$

Can there be a subset $A$ of $ \mathbb R^p$ such that $A^o = \phi$ and $A^- = \mathbb R^p$ where $A^o$ refers to the interior of $A$ and $A^-$ refers to closure of $A$ Attempt: By definition : $(i) ...
4
votes
1answer
101 views

An Integral Inequality

Let $f$ and $g$ be real functions such that $\int_0^\infty(f(x))^2dx<\infty$ and $\int_0^\infty(g(x)^2dx<\infty$. Prove that: $$\left(\int_0^\infty\int_0^\infty\frac{f(x)g(y)}{x+y}dxdy ...
1
vote
1answer
35 views

A fundamental inequality.

I have read in a textbook($p\ge1 ,X_k $ are r.v.): $$\|\sum_{n=0}^k|X_n|\|_p\le(k+1)\|X_k\|_p\quad\text{where}\mathbb E|X_n|^p\le\mathbb E|X_k|^p \quad\text{for} \quad 0\le n\le k $$ I tried to ...
0
votes
0answers
31 views

Some neigbourhood properties

I am given an exercise where I am supposed to show some properties of neighbourhoods. I am given a metric space ($X$,$d$) and two points, $p$ and $q$, that both are members of $X$ ($p$,$q$ $\in$ ...
2
votes
1answer
44 views

Problem showing that $\partial D = \emptyset$

I am considering $(X,d)$, which is a set with a discrete metric and $D \subset X$. With these conditions given, I am supposed to show that $\partial D = \emptyset $. Further I am supposed to describe ...
5
votes
5answers
613 views

Prove that $\lim\limits_{x \to 0} \sinh(x)/x =1$. [on hold]

Prove that $ \lim\limits_{x \to 0} \frac{\sinh x}{x} =1.$ I am having some trouble proving this without derivative. Some help would be much appreciate!
4
votes
1answer
60 views

Prove $f$ is differentiable when $h\to 0$ in definition along rationals.

Let $f:\Bbb{R}\to \Bbb{R}$ be continuous such that for some $x_o\in \Bbb{R}$, $$\lim_{h\to 0,h\in \Bbb{Q}} \frac{f(x_o+h)-f(x_o)}{h}$$ exists and is finite.Prove $f$ is differentiable at $x_o$ I ...
1
vote
3answers
229 views

If the closure of a set $A$ is defined as the intersection of all closed sets which contain $A$, prove that closure of a closed set $B$ is $B$ itself

If the closure of a set $A$ is defined as the intersection of all closed sets which contain $A$, prove that closure of a closed set $B$ is $B$ itself. Attempt: I apologize if this is too basic but I ...
13
votes
1answer
564 views

Integral$\int_1^\infty \log \log \left(x\right)\frac{dx}{1-x+x^2}=\frac{2\pi}{\sqrt 3}\left(\frac{5}{6}\log (2\pi)-\log \Gamma \frac{1}{6}\right)$

UPDATED Hi I am trying to prove the following$$ I:=\int_1^\infty \log \log \left(x\right)\frac{dx}{1-x+x^2}=\frac{2\pi}{\sqrt 3}\left(\frac{5}{6}\log (2\pi)-\log \Gamma \big(\frac{1}{6}\big)\right). ...
2
votes
1answer
38 views

If $X$ is a LCHS and $K, O \subseteq X$ with $K$ cpt & $O$ open, then $\exists U$ open s.t. $K \subseteq U \subseteq \overline{U} \subseteq O$?

I'm having trouble fully understanding the proof of this statement. Suppose $X$ is a locally compact Hausdorff topological space. Then if $K$ is a compact subset of $X$ and $O$ is any open subset ...
2
votes
1answer
109 views

Book recommendations for these types of math?

I'm planning to write a math olympiad in a couple of months (4-5), and am just really trying to get the preparation in. I'm a fairly good math student (did ok in math, not an A+, but I got an A so my ...
0
votes
2answers
85 views

Let $d(a,X)=\inf\{|x-a| / x \in X\}$ then $d(a,X)=0$ iff $a \in \bar{X}$

I've seen answers to similar questions in SE but I need just this How to show $d(x,A)=0$ iff $x$ is in the closure of $A$? In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and ...
3
votes
1answer
161 views

Convergence of $\prod\limits_{n=0}^\infty ( 1+u_{n}) $ when $\sum\limits_{n=0}^{\infty }u_{n}$ and $\sum \limits_{n=0}^\infty u_{n}^{2}$ diverge

I am trying to show that if $u_{0}=u_{1}=u_{2}=0$, and if, when $n>1$, $$u_{2n-1}=\dfrac {-1} {\sqrt {n}}, u_{2n}=\dfrac {1} {\sqrt {n}}+\dfrac {1} {n}+\dfrac {1} {n\sqrt {n}}$$ then $\prod ...
-1
votes
1answer
52 views

Example of a function which is non-lipschitz but satisfies some weaker notion of linear growth

What is an example of a function which is not lipschitz but satisfies the following weaker notion of linear growth $$f(x) < K(1+x) \forall x, K > 0$$ along with being continuous
2
votes
2answers
128 views

Find $\lim_{n \to \infty} \frac{x_n}{x_1 + \cdots + x_n}$

Suppose that $\{x_n\}_{n=1}^{\infty}$ is a bounded sequence, and that $x_n>0$ holds for all positive integer $n$. Find $\lim_{n \to \infty} \frac{x_n}{x_1 + \cdots + x_n}$.
2
votes
2answers
96 views

A matrix $G$ with all eigenvalues with nonzero real part. Then $t\mapsto |\exp(tG)x |$ is unbounded

I am trying to see why this is true. A book I am reading has this claim without any verification and I'm trying to see why it is true. Let $G$ be an $n\times n$ matrix all of whose eigenvalues have ...
0
votes
1answer
215 views

Every uniformly continuous real function has at most linear growth at infinity

Assuming $f:\mathbb R\to\mathbb R $ be an uniform continuous function, how to prove $$\exists a,b\in \mathbb R^+ \quad \text{such that}\quad |f(x)|\le a|x|+b.$$
0
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1answer
43 views

Rudin Priciples of mathematical analysis -P316-Theorem 11.24

Given $\epsilon >0$ We can choose a measurable function s such that $0\leq s \leq f$, and such that $$ \int_{A_1} sd\mu \geq \int_{A_1} fd\mu-\epsilon, \space\space \int_{A_2} sd\mu \geq \int_{A_2} ...
1
vote
1answer
61 views

Prove that these two sets are equal.

Prove that the following two sets are equal: The interval $\left[a,b\right]$ The set $\left\lbrace y\in\mathbb{R}: \text{there exists } s,t\in\left[0,1\right] \text{ such that } s+t=1 \text{ and } ...
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vote
0answers
30 views

net of indicator functions

Hi everyone I was reading Dudley's book and I'm having problems with the following. If $X$ is uncountable, show that there is a net of indicator functions of finite set converging pointwise to the ...
3
votes
1answer
71 views

Proof about sequences of functions.

Is this proof correct? If $\{f_{n}\}$ is a sequence of functions in $C(X,Y)$, $X$ compact, $Y$ complete, and the sequence converges, to $f$, then $K=(\bigcup\{f_n\})\cup \{f\}$ is closed. Proof. ...
1
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1answer
144 views

Every open sub set of $\mathbb R^p$ is the union of countable collection of closed sets

Every open sub set of $\mathbb R^p$ is the union of countable collection of closed sets My textbook gave me hints as follows : Let $G$ be an open subset of $\mathbb R^p$. Let $A$ be the subset of ...
1
vote
1answer
31 views

(a) Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ R$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ that is also ...
2
votes
1answer
39 views

Sequence approaching 1 , proof

Prove $$1 + {\pi\over \sqrt{n}}\stackrel{n\to\infty}{\longrightarrow}1$$ Proof: Let $\epsilon > 0$. We need to find a positive integer $N$, such that $n \ge N$. Now $$\left| 1 + ...
2
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1answer
24 views
0
votes
1answer
59 views

Preperation for a test: Show that inf($\frac{1}{n}$)=0 . Please check if what I have is correct

We are given the following definition: If a sequence $(a_n)$ is bounded from below then it has a greatest lower bound for the sequence called a $\textbf{infimum}$. $m=$ infimum of $(a_n)$ if i) ...
1
vote
2answers
166 views

Show that $\int_{\mathbb{R}^n}f_1f_2 …f_n dx_1 …dx_n ≤ (I_1 …I_n)^{1/(n−1)}.$

For $\quad k = 1,2,...n,\quad$ let $\quad\mathbb{R}^k = \mathbb{R},\quad f_k(x_1,...,x_{k−1},x_{k+1},\ldots,x_n)\quad$ be a nonnegative measurable function on $\quad\mathbb{R}_1\times\ldots\times ...
1
vote
1answer
148 views

In $\mathbb R^p$:Every open subset is the union of a countable collection of closed sets & every open set is the countable union of disjoint open sets

Prove/Disprove that : $(i)$ Every open Set in $\mathbb R^p$ can be written as the union of countable number of disjoint open Sets. $(ii)$ Every open subset of $\mathbb R^p$ is the union of a ...
4
votes
3answers
51 views

About the domain of points having tangent to a curve

Let the graph of $y=f(x)$ be a curve $C$ and $f''>0$. Prove that if $y_0\leq f(x_0)$ then there exist a tangent of $C$ go through $(x_0,y_0)$ I don't know how to prove the existence.
0
votes
0answers
52 views

“Big-O” notation with a Taylor Series Expansion

Use a Taylor's expansion to rid the expression $1-\cos x$ of subtractive cancellation for $x$ small. Use a $\mathcal{O}(x^5)$ approximate. I understand Taylor series and I know that the expansion of ...
6
votes
2answers
750 views

Organizing types of functions by their calculus-related properties, in diagram form?

Does anyone know of a diagram that displays and organizes categories of functions according to their calculus-related properties (e.g. continuous, $C^\infty$, degrees of differentiability and ...
3
votes
0answers
56 views

Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

Let $I = [a, b], E \subset I, m(E) = 0$ (but $E$ not empty). Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$. I am ...