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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Axiomatic Bargaining: Nash's Solution

The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc. Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:. Two individuals can ...
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2answers
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If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$

True or False: If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$ Then: a) $\sum_{n=0}^\infty c_n 2^n$ converges. b) $\sum_{n=0}^\infty c_n 3^n$ converges. Can someone please give me any ...
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0answers
9 views

Fourier transform (properties)

I have a function $f$ such that $|f(x)|\leq e^{-x^2/2}$ hence in $L^2(\mathbb{R})$ and we can compute the Fourier transform $$\hat{f} (\xi) = \frac{1}{\sqrt{2\pi}}\int_{\mathbb R} f(x) e^{-i (u,x)} ...
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14 views

A simple question in calculus (equivalence of limits).

So I want to prove the next equivalence: where D-lim, is $\lim_{n\rightarrow \infty , \ n \notin M \subset \mathbb{N}}$. The easy part, mainly $\Rightarrow$ I did I think good. I am having ...
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Prove this inequality.

Let n $\in\mathbb{N} $ and $x_0,x_1,.....,x_n $ $x_i\in\mathbb{R}$ ,$x_i>0 $ so that $ x_0 + x_1 + .... x_n =1$. Prove that for all $a\in\mathbb{R}$ and $a>0$ the inequality is verified : $\ ...
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1answer
25 views

Prove that f(x) is regulated.

Define $f:[0,1]\to \mathbb{R}$, $f(x):=0$ if $x\notin \mathbb{Q}$, $f(p/q):=1/q$, $q>0$, $p, q$ coprime integers. Prove that $f$ is regulated. A function $f:[a,b]\to\Bbb R$ is a regulated ...
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1answer
27 views

Can we prove the Riemann-Lebesgue lemma by using the Weierstrass approximation theorem?

I'd like to prove the following version of the Riemann-Lebesgue lemma: Let $f: [0,1] \to \mathbb R$ be continuous. Then $$\int_0^1 f(x)\sin(nx) \, dx \xrightarrow{n \to \infty} 0$$ It's quite ...
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1answer
38 views

Given $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , what is $f(0) $?

Let $f:\mathbb R \to \mathbb R$ be such that $f(x)=x+\int_{0}^1 t(t+x)f(t) dt $ , then how do we find $f(0) $ ?
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1answer
20 views

Real Analysis Cardinality Proof

I am trying to prove that given a finite set A and an uncountable set B, A and A union B have the same cardinality. I looks to me like I need to show that there is are injective functions from A to A ...
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2answers
73 views

Convergence of series. Does the hint help?

I need to find value for $x$ for which the series converges, $$\sum_{k=1}^\infty \frac{k^k}{k!}x^k$$ Can I use the following fact? $$\lim_{k \rightarrow \infty} \frac{k!}{\sqrt{2\pi k} ...
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26 views

If $f''(x)$: conti- on $[a,b]$, then there exist $c$ ($c\in[a,b]$) s.t $\int_{a}^{b}f(x)dx=\frac{b-a}{2}(f(b)+f(a))-\frac{(b-a)^3}{12}f''(c)$

I would appreciate if somebody could help me with the following problem Q: Proof that If $f''(x)$: conti- on $[a,b]$, then there exist $c$ ($c\in[a,b]$) s.t ...
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How is the interchange of the limit and the maximum valid at this point in Erwin Kreyszig?

In 1.5-5 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, the author shows completeness of the space $C[a,b]$ of all (real- or complex-valued) functions defined and continuous ...
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2answers
30 views

Whether limit of a function exists

Let $f:[0,\infty) \rightarrow \mathbb R$ be a function such that for any positive $a$ the sequence $f(an)$ converges to zero.Does the limit $lim_{x\rightarrow \infty} f(x)$ exist? My problem is ...
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2answers
21 views

Continuous differentiability of atan2

Consider the function atan2 defined on the plane, minus the origin and the negative $x$-axis, as the unique $\theta$ such that $-\pi<\theta<\pi$ and $$ x = r \cos \theta, \qquad y=r \sin ...
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1answer
18 views

$F(x)=\int_{a}^x f(t) dt$ is uniformly continuous where $f\in C[a,b]$

Prove that $F(x)=\int_{a}^x f(t) dt$ is uniformly continuous, where $f\in C[a,b]$ Can I say here $f(t): t\in [a,b]$ is bounded? Does this $F(x)=\int_{a}^x f(t) dt $, imply $f$ is bounded? What ...
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4answers
46 views

Supremum of a Continuous Function is Continuous

I'm working on this problem from Elementary Analysis by Ross which is intuitive when sketched but keeps stymieing me when I try to write it out. Let $f$ be a continuous function on $[a,b] \subset ...
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2answers
20 views

Convergence of a series for particular values of x

I am having a problem proving for which values of $x \in R$ does $$\sum_{n=1}^\infty \frac{(-1)^n}{x+n}$$ converges? Is the convergence uniform on $(-1,1)$? Doing the root test I arrive to $\alpha ...
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22 views

Nowhere dense set and boundary points

If $int(\bar{X}) = \emptyset$ and $X$ not empty, does it mean all points of $X$ are boundary points? I think it does. My reason is the following: Since $X \subset \bar{X}$, we know $int(X) \subset ...
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1answer
40 views

prove or show that

If $x,y \in\mathbb R$ with $x < y$, show that $x \leq ty+(1-t)x \leq y$ for all $t \in [0,1]$. I have so far is: any point in ( x , y ) can be given by x + multiple of [ y - x ] if the multiple ...
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2answers
14 views

$\int_{x}^{x+1} |f(y)| dy \le C$ with an $C > 0$ for all $x > 0$, prove that $\sup\limits_{0 < y} |f(y)|$ exists

It is $\int_{x}^{x+1} |f(y)| dy \le C$ with an $C > 0$ for all $x > 0$. It is obvious that $\sup\limits_{0 < y} |f(y)|$ exists, but how can I formally prove it? I know that $\int_{x}^{x+1} ...
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Does this bounded sequence converge?

The sequence $(a_n)$ is bounded for $n=1, 2, \dots$, such that $$a_n \leq \frac{1}{2} \left(a_{n-1} + a_{n+1}\right)$$ for $n \geq 2$. I want to prove the sequence $(a_n)$ converges. Since I am ...
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2answers
37 views

Convergence and absolute convergence of a series

I would like to prove that the following series converges, but not absolutely, $$\sum_{n=1}^\infty \left[ \exp \left(\frac{(-1)^n}{n} \right) -1\right] $$ Since no other information is given, I have ...
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1answer
34 views

On compact set in real analysis

Let K be a compact set. Prove that $\forall \, \epsilon > 0$, K can be covered with a finite number of neighbourhoods of radius $\varepsilon$. Show that the reciprocal is not true. My approach. ...
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3answers
28 views

Convergence of single variable power series

I would like to know whether the following power series converges or diverges. $$1-x+\frac{x^2}{2!}-\frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.$$ My intutition tells me that for any nonzero $x$, the ...
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1answer
23 views

Dirichlet Problem

I have to show the Dirichlet Problem for real-value $$u \in C^2(D)\cap C(\overline{D})$$. $$\Delta u=0$$ $$u(e^{it})=1/2(e^{it}+e^{-it})$$ Now should I calculate Laplacian of u than to equal it with ...
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23 views

Differential inequality involving derivatives

I'm having trouble with a differential inequality. Consider a smooth function $f(x)$ defined for $x>0$ with $f'>0$. Given $0< a < b$, show that there exists smooth functions $g(x)$ and ...
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11 views

$\int\limits_{x}^{x+1} |f(x-\frac{z}{a})| dz \le K$ with $a < 0$ and $0 < x+1 < -a$ and a $K > 0$ for every $x > 0$

It is $\int\limits_{x}^{x+1} |f(y)| dy \le C$ for a $C > 0$ and every $x > 0$. Is that enough to prove the following? $\int\limits_{x}^{x+1} |f(x-\frac{z}{a})| dz \le K$ with $a < 0$ and ...
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1answer
26 views

Help on proof of claim about $\limsup$ of real sequence.

Let $\{p_n\}$ be a sequence of real numbers. I would like to show that $~lim_{n \rightarrow \infty}\sup~p_n< \infty$ if and only if $\{p_n\}$ is bounded above. Let $E:=\{\text{limits of ...
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$ l^{\infty}$ is not compact.

Prove that $l^{\infty}$ is not compact. My idea is: Let $ X=l^{\infty}$ defined by $ \{e^n=(0 ,0 .....\underset{nth}{1},0.......)\}$ This set is bounded and closed both but not totally bounded.(has ...
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1answer
71 views

Does $1-1+1/2-1/2+1/3-1/3+\cdots$ converges?

Is this true the the above series converge? What might be a economic way to show this? I tried to rearrange them but it seems like they are not all positive so its a bit dangerous. So it is ...
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1answer
16 views

Real 2D Analysis Question using Brouwer's Fixed Point Theorem

My question is as above. Currently I am stuck at the very start, part (i)! I can't come up with an appropriate $f$, even though I've been thinking about it for ages! If someone would be able to ...
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21 views

Proof check of sum of a compact and closed set of real numbers is closed

Let $A$ be a closed and $B$ be a closed and bounded set in $\mathbb R$ , then we have to show that $A+B:=\{a+b:a\in A , b\in B \}$ is closed in $\mathbb R$ . My Proof : Let $\{a_n+b_n\}$ be a ...
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Does $\sum\limits_{k=1}^{\infty} (a_{k+1} - a_k)^2 <\infty$ imply $\frac{1}{n^2} \sum\limits_{k=1}^n a_k^2 \to 0$?

I'm currently working on some problem regarding Dirichlet forms and, as the title states, I'm trying to figure out if $$\sum\limits_{k=1}^{\infty} (a_{k+1} - a_k)^2 <\infty \Rightarrow ...
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46 views

Is $\mathbb{R}^n$ not nowhere dense? [on hold]

Is $\mathbb{R}^n$ not nowhere dense? I am trying to show that a clopen set is not necessarily nowhere dense. I know that it is both open and closed, but I am not sure how to find the interior of ...
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29 views

Taylor Series Expansion for Function of Two Variables (with Countable Discontinuities)

Given a real-valued function of two real variables, under certain conditions of smoothness in a closed ball about some point, we can obtain a Taylor series for the function about that point. I want ...
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1answer
33 views

Measurability of $\{(x,y): x\in M,0\leq y\leq f(x)\}$

Let $(X,\mathfrak{S}_x,\mu_x)$ be a measure space endowed with the $\sigma$-additive and complete measure $\mu_x$ defined on the $\sigma$-algebra $\mathfrak{S}_x$, let $\mu_y$ be the linear Lebesgue ...
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Proving one limit is equal to another

Let ($x_s$) be a convergent sequence, where $x_s>0$ for all s, and $y_s$ be a sequence such that $$\displaystyle \Large y_s=\frac{s}{\frac{1}{x_1}+...+\frac{1}{x_s}}$$ Prove $\displaystyle \lim ...
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What is $\int_0^{\infty} x^2e^{\frac{(x-\mu)^2}{2 a^2}} dx$?

How can we express the integral $\int_0^{\infty} x^2e^{-\frac{(x-\mu)^2}{2 a^2}} dx$ for example by means of the error function? The problem is of course, that the expectation value is shifted and we ...
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2answers
47 views

Is this sequence decreasing?

If a sequence $b_n>0$ and $b_n$ converges to $0$, can we say it is eventually decreasing? This problem bumps up when I am trying to something bigger. However, I am very unsure of this. If this is ...
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102 views

Test for convergence $\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$

What is the easiest way to test the convergence of $$\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$$ Is it possible to only use the high school tools for that?
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1answer
21 views

If $y$ is the solution of $\left\{y'=-y+\sqrt{t},y(0)=y_0>0\right\}$, then $\lim_{t\to\infty}\frac{y(t)}{\sqrt{t}}=1$

The homogeneous equation $$y'=-y$$ has the solution $$y_h(t)=ce^{-t}\;\;\;\;\;(c,t\in\mathbb{R})$$ In order to find a particular solution we can take the approach $$y_p(t)\stackrel{!}{=}c(t)e^{-t}$$ ...
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1answer
35 views

To show $(x+y)^p\leq x^p+y^p$, where $0\leq p\leq1, x>0,y>0$? [duplicate]

How to show that, $(x+y)^p\leq x^p+y^p$, where for $0\leq p\leq 1,x\geq 0, y\geq0?$ Any suggestion how to prove it? Thanks in advance.
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11 views

Area under mapping

Can some one help me with this problem: Let $f:R^2\rightarrow R^2$ be defined by $\displaystyle{f(x,y)=(e^{x+y},e^{x-y})}$ Find the area of the image of the region $\{(x,y) \in R^2 : 0<x,y<1 ...
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2answers
28 views

Convergence of cosine

Does the following sequence $$(\cos (\pi \sqrt{ n^2 + n}))_{n=1}^\infty $$ converge? I was trying the ratio or root test, but they don't seem to work in this case. Mean value theorem?!
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5answers
156 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
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1answer
34 views

Particular case of every sequence has a Cauchy subsequence?

A metric space (X,d) has the following property: Given $\epsilon >0$ and non-empty finite subset $X_\epsilon \subset X$ $$ \inf \{ d(x,p) : p \in X_\epsilon \} < \epsilon$$ for $x \in X$ I ...
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0answers
29 views

question in Numerical analysis

please guide me how to start and I will continue the another steps
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37 views

A question about $f(x)\equiv C$

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
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1answer
22 views

Help clearing up the definition of Limsup?

I was thinking about the equivalence of the two following definition of Limsup of a sequence. I find the definition 1 much more intuitive and I have been trying to convince myself of the equivalence ...
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0answers
8 views

Integral over homogeneous function does not vanish

Let $\alpha>0$ be a multi-index. For $x,y\in\mathbb{R}^n$, $n>1$, we consider the integral $$\int_{|x|=1} \int_{|y|=1} \partial_y^\alpha f(y)\ g(x,y)\ \mathrm{d}y \mathrm{d}x\qquad (*)$$ where ...