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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10 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 ...
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1answer
23 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} ...
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2answers
89 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 $\lim_{n \to \infty} \frac{x_n}{x_1 + \cdots + x_n}$.
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39 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 ...
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14 views

Proving that $F_{x,n} = \left\{y \in \mathbb R^p :|y-x| \leq \dfrac {1}{n}\right\}$ is contained in $G$

Let $G$ be an open subset of $\mathbb R^p$. Let $A$ be the subset of $G$ whose coordinates are all rational numbers. Then, show that For each $x$ in $A$, there is a smallest natural number ...
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1answer
21 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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0answers
19 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 ...
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1answer
18 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
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1answer
31 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 + ...
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1answer
50 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 ...
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0answers
17 views

Showing a sequence is a Cauchy sequence in $\operatorname{End}(\Bbb R^n)$

Let $A \in \operatorname{End}(\Bbb R^n)$ with $\| A \| < 1$. Show that $( \sum_{0 \leq k \leq l} A^k )_{l \in \Bbb N}$ is a Cauchy sequence in $\operatorname{End}(\Bbb R^n)$, and conclude that ...
2
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1answer
16 views

Existence of an exponential double integral (for the probabilists: Are the $L^p$-norms of Brownian local time integrable in the space variable?)

I have encountered the following integral and, with a lot of handwaving and some identities for Gaussian integrals (see for example ...
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1answer
36 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) ...
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1answer
28 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 ...
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13 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 ...
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0answers
38 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 ...
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1answer
25 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
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19 views

integration formula

help me please all these functions are regular. How we can found this formulation $$ \displaystyle\int_{\Omega} (f(u)-f(k)) \nabla p(g(u)-g(k)) \xi dx = - \displaystyle\int_{\Omega} H(u,k) \nabla \xi ...
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4answers
90 views

Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24} \end{equation} I tried to use by parts method and ended with \begin{equation} \int \ln^2(\cos ...
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23 views

Prove that the set of all infinite sequences of positive integers has the power of the continuum.

Prove that the set of all infinite sequences of positive integers has the power of the continuum. By definition, the positive integers are defined to be $\displaystyle\aleph_{0}$ in terms of ...
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53 views

How prove $f(a_{i})=0$ if $\int_{0}^{1}x^kf(x)dx=0,k=1,2,3,\cdots,n$ [duplicate]

let $f(x)$ is continuous on $[a,b]$,and such $$\int_{0}^{1}x^kf(x)dx=0,k=0,1,2,3,\cdots,n$$ show that: there exsit $n+1$ different $a_{1},a_{2},\cdots,a_{n},a_{n+1}(a_{i}\neq a_{j},\forall ...
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1answer
35 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
40 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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35 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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21 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
55 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
38 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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32 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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2answers
70 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
31 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
30 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
38 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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28 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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0answers
9 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
25 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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42 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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24 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
45 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
81 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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1answer
52 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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2answers
39 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
39 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
117 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
40 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 ...
5
votes
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 ...
0
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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 ...