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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28 views

Contradiction in an Alternative Definition of an Open Set?

A set $G$ in $\mathbb R^p$ is said to be open in $\mathbb R^p$ if , $\forall x \in G$, $\exists r \in \mathbb R^+$ such that every point $y$ in $\mathbb R^p$ satisfying $|x-y|<r$ also belongs to ...
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1answer
32 views

Does $f_n(a_n)\to f(a)$ hold?

Say, we have $f, f_n \in C^0(\mathbb R, \mathbb C)$ such that $f_n \xrightarrow{\text{uniform}}f$ and a sequence of reals $a_n \to a$. Does it then hold that $f_n(a_n)\to f(a)$? I couldn't think of ...
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27 views

Error? An open subset of $\mathbb R^p$ is connected if and only if it can be expressed as the union of two disjoint non-empty open sets.

I believe the book which I am reading has a printing error. One of the lemmas reads like this An open subset of $\mathbb R^p$ is connected $\iff$if it can be expressed as the union of two disjoint ...
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19 views

Boundary, closure, interior [on hold]

$X=(0,4] \cup \{6\} \cup[10,11]$ is subspace of $\mathbf{R}$. If A is $A=(0,2] \cup \{6\} \cup(10,11]$, find $IntA$, $ClA$, $FrA$ in subspace $X$, where is: Int - interior, Cl -closure, Fr-boundary. ...
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1answer
33 views

Misunderstanding about Laplace operator

Let $\Omega$ be a bounded subset of $\mathbb{R}^n$. We know that the Laplace operator \begin{align} \Delta \colon H_0^1(\Omega) \to L^2(\Omega) \end{align} admits an inverse operator \begin{align} A ...
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2answers
34 views

Using the mean value theorem to prove inequalities

Using the Mean Value Theorem, show that for any $t>0$, $$\left|e^{-x^2/2t}-e^{-y ^2/2t}\right|\leqslant \frac{|x-y|}{t}$$ for all $x,y$ with $|x|,|y|\leqslant 1$. My attempt. Without loss of ...
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0answers
28 views

Expected values of continuous and bounded functions are equal then random variables are equal, too.

I have seen several of reasoning based on the following fact: Real random variables $X, Y$ in $\mathbb{R}^n$ are equal almost surely if and only if $\mathbb{E}g(X)f(X) = \mathbb{E} g(X)f(Y)$ for ...
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1answer
55 views

prove that $ a^2 + b^2 + c^2 \ge 2\left( {a^3 b^3 + a^3 c^3 + c^3 b^3 + 4a^2 b^2 c^2 } \right)$

I have: let $$a;b;c$$ be non-negative real numbres with sum 2.prove that $$a^2 + b^2 + c^2 \ge 2\left( {a^3 b^3 + a^3 c^3 + c^3 b^3 + 4a^2 b^2 c^2 } \right)$$ I should determine whether this ...
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1answer
29 views

Continuity of f [on hold]

Is the statement below true.If it is could someone provide a proof of this.If its not provide a counter example $ f(x)$ is continuous at $x_0$ $\implies \exists \delta>0:$ (if ...
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1answer
28 views

Find the coefficients $a,b$ so that $\|\vec x\|=a(|x_1|+|x_2|)+b\max(|x_1|,|x_2|)$ is a norm

For which coefficients $a,b$, the expression: $$\|\vec x\|=a(|x_1|+|x_2|)+b\max(|x_1|,|x_2|)$$ is a norm in $\mathbb R^2$? My attempt: I need to verify the properties of the norm: Triangle ...
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1answer
37 views

$\{X_n\}$ is a bounded, divergent, infinite sequence of real numbers, prove [on hold]

prove (A) $\{X_n\}$ contains convergent subsequences with different limits. (B) $\{Y_n = \min_{k\le n} X_k\}$ is convergent. not sure if (A) is correct.
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23 views

Disconnected Sets definition and connectedness of the unit interval

The definition of a disconnected set seems a bit ambiguous in the book I am reading : $1.$ A subset $D$ of $\mathbb R^p$ is said to be disconnected if there exist two open sets $A$ and $B$ such ...
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6 views

Description of distributions with support in a linear subspace

The following lemma is true: any distribution $\lambda$ on the real line with support included in $\{0\}$ can be written as $$ \lambda = \sum_{i = 0}^N a_i \partial^i(\delta_0)$$ with the $a_i$ being ...
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1answer
24 views

How I can imply that the supremum is in a set S?

I want to prove this: Let $S$ be a nonempty set of real numbers that ins bounded from above (below) and let $x=sup S(infS)$.Prove that either $x$ belongs to $S$ or $x$ is an accumulation point of ...
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20 views

multipication operator derivative

let $F:W\to W $ be a function. $W=C(\Omega ) , \Omega .$ is bounded set in $R^n$ I try to understand how frechet derivative operates on arbitrary function h. if for example $ F[u]=u^2 $ then can ...
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41 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 $8x\ge13, 8y \ge 13$?
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22 views

LUB, GLB, maximum and minimum of a set .

I am not sure if my solution for the following problem is correct. Evaluate LUB, GLB, maximum and minimum (if they exist) of $\{-n: n ∈ \Bbb N\}$. My answers: LUB: $-1$ GLB: $-\infty$ max: $-1$ ...
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1answer
44 views

Parallel vectors in $\mathbb{R}^n$.

Def: We say that $\vec{x},\vec{y}\in\mathbb{R}^n$ are parallel vectors if $|\vec{x}\cdot \vec{y}|=||\vec{x}||\,| |\vec{y}||$. (i.e equality holds in Cauchy–Schwarz inequality) I'm having some ...
3
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1answer
70 views

Minimum of the function $f(x)=\frac{1}{1+x^2}+\frac{3}{1+(h-x)^2}$, for $0\leq x \leq h$

Find the minimum of the function $f(x)=\frac{1}{1+x^2}+\frac{3}{1+(h-x)^2}$, for $0\leq x \leq h$ Proof that the solution can be expressed as: 1. There exists a $δ>0$ so that for $0\leq h \leq ...
5
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49 views

Convergence of Integral near 0

I am trying to determine the convergence of the integral \begin{equation} \int_0^1 \frac{f(x)}{x}\, dx \end{equation} given that $f(x)$ is bounded and continuous on $[0,1]$, and that $f(x)=0$. The ...
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1answer
40 views

showing these sets are countable

Prove that the following sets are countable : Numbers like $0.80$ and $0.6999...$ (2 decimal expansions) Intervals with rational endpoints The set of all rational points in the plane. For the ...
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2answers
35 views

Basic sequence question

Let $\{x_n\}$ be a sequence of real numbers. If $x_n\geq 0$ $\forall$ $n\in \mathbb{N}$, show that $\displaystyle x=\lim_{n\rightarrow \infty} x_n \geq 0$. I know that this is quite an easy problem ...
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1answer
38 views

$f$ is continuous on $X$ iff $f$ is continuous on every compact subset

Let $(X,d)$ be a metric space,then prove that $f$ is continuous on $X$ iff $f$ is continuous on every compact subset of $X$. If $f$ is continuous on $X$ then its restriction on each compact ...
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1answer
15 views

Relation between metric and uniform convergence

I have a question about a relation between metric and uniform convergence on $\mathbb{R}$ Question It is true that the extended real $\overline{\mathbb{R}}$ is completely metrizable. Let ...
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3answers
116 views

How to evaluate $\int{d(y^2)}$?

Can anybody help me to solve this integral please: $$\int{dy^2}$$ Here $dy^2$ means $d(y^2)$, not $(dy)^2$. Thanks for any help.
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1answer
36 views

Question about limit points in relation with continuity and functional limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I have the feeling that the author is being careless about limit points in his theorems or I am not understanding ...
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1answer
114 views

If $f(x+y)=f(x)+f(y)$ and $f$ is monotone, then prove $f(x)=ax$

Suppose $f:\Bbb{R}\to\Bbb{R}$ is a monotone function function satisfying $$f(x+y)=f(x)+f(y) \quad \forall x,y\in\Bbb{R}$$ then prove $f(x)=ax \quad \forall x\in\Bbb{R}$ I proved that $f(x)=ax ...
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1answer
23 views

O Notation and taylor series

Wolframalpha tells me that the Taylor series of the exponential function is $1 + x + \frac{x^2}{2}+ O(x^3).$ Taylor series I just don't get this big O there, shouldn't this be a small o?
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1answer
26 views

additive subgroup of real numbers with non empty interior

G is an additive subgroup of real numbers with a nonempty interior.Then G is all the real numbers.what is the exact proof?
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2answers
67 views

A continuous mapping $f:\mathbb{R}\rightarrow\mathbb{R}$ may have a fixed point?

Let a function $f:\mathbb{R}\rightarrow\mathbb{R}, $satisfied $$\forall x,y\in\mathbb{R},|f(x)-f(y)|\leq k|x-y|.(0<k<1)$$ Prove: There exists a only one $\xi\in \mathbb{R}$ ,such that ...
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31 views

How to prove $e^x \ge x^e$ for all $x \in \mathbb{R}$ [on hold]

I need help with how to prove these two equalities hold true? $$e^x \ge x^e \text{ for all } x \in \mathbb{R}$$
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32 views

to prove that a metric space which is not complete [on hold]

Given $C_{\infty}= \{x=(x_n) :x_n \in \mathbb{R}\ \text{and}\  \exists\ n(x)\in \mathbb{N} $ s.t $x_n = 0\ \forall\ n > n(x)\}$  Where each element is sequence of type ...
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1answer
13 views

Proof of FTC, continuity part, for Lebesgue integrable functions

The part of the FTC I am interested in says: If $f$ is a Lebesgue-integrable function on $[a,b]$, then $F(x)=\int_a^xf(t)\,dt$ is continuous. This is usually considered a lemma or something for ...
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1answer
57 views

Clarification needed: $\inf (A+B) = \inf A + \inf B$

Let $A$ and $B$ be nonempty bounded subsets of $R$ and let $A+B$ be the set of all sums $a+b$ where $a\in A$ and $b\in B$. Prove $\inf(A+B)=\inf(A)+\inf(B)$ My attempt: Since $A$ and $B$ are ...
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3answers
41 views

Correctness of proof that every positive rational with square $>2$ is an upper bound for those with square $<2$

I would like to know whether my proof makes sense or not, and if not where should it be corrected. Let $E=\{x \text{ is rational }: x>0 \text{ and } x^2<2\}.$ Claim: Every member of $F=\{x ...
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2answers
43 views

Correctness of the proof that the set $\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$ does not have a smallest element

Let $F=\{x \in \mathbb{Q} : x>0 \text{ and } x^2>2\}$. I am asked to show that $F$ does not have a smallest element. The hint is to simply prove the claim: 'If $p$ is a rational number in ...
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55 views

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, …,$ on the interval $[0,1].$

Consider the sequence $f_n(x) = (\sin(πnx))^n , n = 1, 2, ...,$ on the interval $[0,1].$ Prove that for any $δ > 0$ there is a set $E ⊂ [0,1]$ with $m(E) > 1−δ,$ and a subsequence $f_{n_k} (x), ...
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1answer
50 views

$|\{0,1\}^\infty| = |\mathbb{R}|$?

Let $\{0,1\}^\infty$ = {$(a_n)_{n \in \mathbb{N}}; a_n \in \{0,1\} \forall n \in \mathbb{N}$} Is there a bijection between $\{0,1\}^\infty$ and $\mathbb{R}$? I thought about something like this: If ...
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2answers
75 views

Characterization of continuity in terms of preimages of open sets

1--8 Theorem. If $A\subset \mathbb R^n$, a function $f:A\to \mathbb R^m$ is continuous if and only if for every open set $U\subset \mathbb R^m$ there is some open set $V\subset \mathbb R^n$ such ...
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1answer
45 views

Differentiability of the remainder in Taylor's theorem

Suppose we have a function that's differentiable $m$ times over $[a,b]$, we have $a< \alpha < x < b$ and $n < m$. Then $$ f(x) = \sum_{i = 0}^{n-1} \frac{f^{(i)}(\alpha)}{i!}(x - ...
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1answer
35 views

In the definition of sequences diverging to infinity, why must the constants be positive?

We are given the following definitions A sequence $(a_n)$ diverges to $\infty$ if for each $ M \in \mathbb{R}^+ \exists N_M \in \mathbb{N} \ \text{such that } \\ a_n > M \ \forall n \geq N_M$ ...
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24 views

Measure of Elementary Sets Proof

I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the ...
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94 views

A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

I am looking for a closed-form of this integral \begin{equation} \frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx \end{equation} I ...
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1answer
42 views

Showing two functions are uniformly continuous

I have no idea how to prove this detail (uniformly continuous) about these functions because they're defined to $\infty$. I need the general mindset to prove it, or any ideas. Thanks in advance. $$ ...
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95 views

Fractional Part of $ a^n $

Prove that there exists a real number $ a>1 $, such that $ \{a^n\} $ belongs to $[\frac{1}{3},\frac{2}{3}]$ for all positive integers $n$ and $\lfloor a^n\rfloor$ is even iff $n$ is a prime. ...
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2answers
23 views

Proof of Divergence Criterion for Functional Limits

I'm self-studying from the book Understanding Analysis by Stephen Abbott, and I don't understand corollary 4.2.5 on page 107. To be more specific, let me first write down the theorem that precedes ...
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1answer
40 views

Axioms of seperation

I am studying topological spaces, and I have seen that there are $3$ main axioms of separation: $\mathrm{T1}$, Hausdorff and normal. Now, between Hausdorff and normal there is a case where: given ...
2
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1answer
56 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 ...
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9 views

Does symmetric decreasing rearrangement of a smooth function preserves smoothness?

Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing rearrangement of ...
3
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6answers
128 views

Limit of sequences: $\lim \frac{(2n)!}{(n!)^2} $

Verify if the sequence $$\frac{(2n)!}{(n!)^2}$$ converges. My attempt: $$\frac{(2n)!}{(n!)^2} = \frac{(2n)(2n-1)...(n+1)}{n.(n-1)...1} \geq \frac{(n+1)^n}{n!} $$ Maybe it is easier to show that ...