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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Prob. 7, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: The dual space of a Hilbert space is a Hilbert space.

Here's Prob. 7, Sec. 3.8 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that the dual space $H^\prime$ of a Hilbert space $H$ is a Hilbert space with inner product ...
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1answer
12 views

Continuous function and its Mapping

Let $f: \mathbb R \to \mathbb R$ be a continuous function. Which one of the following sets cannot be the image of $(0,1]$ under $f$? $\{0\}$ $(0,1)$ $[0,1)$ $[0,1]$. We know that $(0,1]$ is ...
3
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2answers
62 views

Prove this limit $\lim \limits_{x\to\infty}f(x)=0$

I have this problem in real analysis. I think it needs integral factor or knowledge of ODE to prove, but not sure how to it. Here is the question: Let $f$ be a real valued continuous function on ...
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1answer
13 views

Proving a corollary of a corollary of the Mean Value Theorem (corollary-ception)

This is will a wordy question but here it goes: My analysis book states the mean-value theorem and then a corollary which we will label as (1): Let $f$ be a differentiable function on $(a,b)$ such ...
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0answers
12 views

$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n)$

I am trying to prove the equality $$\mathcal B(\mathbb R^{m+n})=\mathcal B(\mathbb R^m) \otimes \mathcal B(\mathbb R^n),$$where $\mathcal B(\mathbb R^i)$ is the Borel $\sigma$-algebra on $\mathbb ...
3
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2answers
39 views

Question on proof of Heine-Borel theorem

Spivak's book on calculus on manifolds has a statement that I can't grasp. Say we have the closed interval $[a,b]\subset\mathbb{R}$ covered by $\mathcal{O}$ and we define $$A=\{x \in [a,b]:[a,x] ...
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0answers
13 views

inverse function theorem for analytic functions whose derivative might vanish

Suppose $x(t), y(t)$ are monotone increasing functions, and $f$ and $g$ are real-analytic functions that are not identically zero. If $f(x(t)) = g(y(t))$ for all $t$, does it follow that $x$ is an ...
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1answer
30 views

if $\lim a_n = \infty$ and $\lim b_n = B$, then $\lim (a_n+b_n) = \infty$

I'm having trouble starting the proof not sure exactly how to go about it. So far I know that for a sequence to go to infinity it means that for all $n >0$ there exists $n_0$ for all $n$ greater ...
3
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1answer
40 views

A function with midpoint-linear derivative is a quadratic polynomial

Suppose $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $$f'\left(\frac{a+b}{2}\right) = \frac{f'(a)+f'(b)}2,\quad \forall a,b\in\mathbb{R}$$ Prove that $f$ is a polynomial of ...
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24 views

An interval covering problem [on hold]

Consider a set $A$ of intervals $[a_i,b_i)$, whose union is $[0,1)$. Prove there must exist a subset $B$ of $A$ such that the intervals in $B$ are pairwise non-overlapped and the sum of their lengths ...
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3answers
46 views

A recursive sequence is defined by…

A sequence is defined recursively by $a_1=1$ and $a_{n+1} = 1 + \frac{1}{1+a_{n}}$. Find the first eight terms of the sequence $a_n$. What do you notice about the odd terms and the even terms? By ...
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0answers
19 views

Convergence in $L^2$? [on hold]

Does anybody see how we can show that $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{1}{|x+ 2 \pi l |^{2m}} \right)^{-\frac{1}{2}}$$ for $x \neq 2 \pi l$ with ...
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0answers
7 views

Comparison between interpolation and Tikhonov regularization.

Interpolation is defined as finding a value of a function between two points and one can think of Tikhonov regularization as to estimate a suitable function under certain condition. Can we think ...
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1answer
32 views

W.Rudin (1.22 decimals)

I am reading the book of Rudin "Principles of mathematical analysis". In 1.22 decimals he wrote: Let $x>0$be real. Let $n_0$ be the largest integer such that $n_0\leqslant x$ (Note that the ...
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3answers
53 views

$f(x)=\sum_{n=0}^{\infty}a_n x^n$ and there exists a sequence $(x_n)$ tending to $0$ such that $f(x_n)=0$ for all $n$, then $f(x)=0$ for all $x$.

I found this question really difficult for me, I don't even know how to start with it? Could you help me? I will appreciate that. Prove that if $f(x)=\sum_{n=0}^{\infty}a_n x^n$ (defined in ...
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1answer
20 views

Show a Function is strictly monotone Increasing, and what does it say about its inverse?

For example: $$g(x)=x^3-3x^2-1 \quad, \quad x\in [2,+\infty]$$ What I have tried to do was to take the first Derivative. I get $$ g'(x)=3x^2-6x$$ I then check the sign of Derivative of g(x) at ...
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2answers
43 views

If a $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how to find sequence $\{b_n\}$ such that $\sum |b_n|<\infty$ but $\sum |a_n||b_n|$ diverges?

If we are given any sequence of real numbers $\{a_n\}$ diverges, so $a_n \rightarrow + \infty$, how can we find a sequence $\{b_n\}$ such that $\sum |b_n|$ converges but $\sum |a_n||b_n|$ diverges? I ...
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1answer
19 views

Existence of a smooth function with given derivative roots

Is there a smooth function $f$ that for all $n\in\mathbb{Z}_+$, $f^{(n)}(n)=0$ i.e. $n$th derivative at the point $n$ is zero and $f^{(n)}(x)\ne 0$ for all $x\in\mathbb R\setminus \{n\}$? If there is ...
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1answer
24 views

Which of two quantities is greater?

Let $x$ and $y$ be two positive real numbers such that $x>y$. Which of the quantities is bigger and when? $(x-y)\log\left(1-\frac{y}{x}\right)$ $x\log\left(1-\frac{y}{x+y}\right)$
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0answers
27 views

Suppose that $f:(0,\infty)\rightarrow \mathbb{R}$ is a function satisfying $f'(x)=1/x$ for all $x\in (0,\infty)$, show $f(xy)=f(x)+f(y)$. [duplicate]

I would like to ask you for some help with the following problem. Suppose that $f:(0,\infty)\rightarrow \mathbb{R}$ is a function satisfying $f'(x)=1/x$ for all $x\in (0,\infty)$, and $f(1)=0$. Show, ...
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1answer
27 views

Construction of a smooth function $\varphi$ such that the sum of $\varphi(2^{-n}t)$ is constant

I want to construct $\varphi \in C_c(\mathbb R)$ whose support is in $[\frac 1 2 ,2]$, satisfying $\displaystyle\sum_{n=-\infty}^\infty \varphi(2^{-n}t)=1$ for all $t>0$. I guess I should use ...
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1answer
17 views

Heat Equation on $[0,l]$ with Neumann boundary conditions

I was reading the following pdf about the heat equation on an interval $[0,l]$ with Neumann conditions, http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_N_web/heat_ex_homo_neum.pdf i.e. ...
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0answers
29 views

Understanding series and their sums

Here's something that I can't wrap my head around while self-studying analysis. Is defining a function to be a series and defining a function to be the sum of a series considered to be two different ...
3
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1answer
31 views

Extreme values of a two-variable polynomial

Is it possible to find a two-variable polynomial which has only two extreme values on the whole plane, one is a local maximum, another is a local minimum, and the local maximum is less than the local ...
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25 views

The $\{ 0,1\}^{\mathbb{N}}$ bit-sequences for the metric space.

Let $M=\{ 0,1\}^{\mathbb{N}}$ denote the set of bit-sequences. For two bit-sequences $x=(x_{n})_{n\in \mathbb{N}}$ and $y=(y_{n})_{n\in \mathbb{N}}$ let $\mu(x,y)=\min\{ n\in \mathbb{N}\mid ...
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2answers
26 views

How to evaluate limits

Let $f$ be a continuously differentiable function on $\mathbb R$. Suppose that $L=\lim\limits_{x\to \infty}(f(x)+f^{'}(x))$ exists. If $0<L<\infty$, and if $\lim\limits_{x\to \infty} f^{'}(x)$ ...
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1answer
31 views

Spivak Ch1 Proof Critiques

I've started working through Spivak's Calculus. I'm going into senior year after this summer, took the AP Calculus BC test last year, and wanted to get a firmer foundation in calculus before I take ...
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4answers
48 views

General expression for the k-th derivative of $(\cos(x))^n$

Is there a general expression for the higher-order derivatives of $(\cos(x))^n$ evaluated at the origin? The odd derivatives are zero due to the symmetry, but what about the even derivatives?
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23 views

Showing sequence function is monotone [duplicate]

Is this sequence of functions monotone? $$f_{n+1}(x)=\varphi(x)f_n(x)+\frac{1}{\varphi(x) f_n(x)}, \forall x \in[0,+\infty)$$ Where $\varphi:[0,+\infty)\to \mathbb{R}$, $1/2\leq \varphi(x) <1$ ...
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0answers
17 views

Non-asymptotic error bound

I am looking for a sufficiently tight estimate of the following over the interval $[-T,T]$: $| \exp(t) - (1+t/n)^n|$. This is of course $o_n(1)$. What I am looking for is a non-asymptotic estimate ...
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1answer
29 views

Is point-to-set distance function $C^\infty$ for $\mathbb{R}^n$

Let $x\in \mathbb{R}^n$ and $Q\subset \mathbb{R}^n$. Then we define the point-to-set distance function as: $$ d_Q(x) = \inf_{y \in Q} \| x-y\| $$ It's continuous for every normal space (not only ...
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0answers
9 views

An optimization problem with a simplex constraint

Suppose $X^i=[0,1]$ for $i=1,2,3$. $X=\prod_i X^i$ and $\mu_i$ is a measure on $B([0,1])$ and $\mu$ is the product measure. Let $f,g,h$ be $L^2(\mu)$ integrable functions satisfying $$0\leq ...
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5answers
87 views

Why is $\operatorname{Int}(A) \cup \operatorname{Int}(B) \neq \operatorname{Int}(A \cup B)$?

I know that $\operatorname{Int}(A) \cup \operatorname{Int}(B) \subset \operatorname{Int}(A \cup B)$, but that the other direction does not hold, so can anybody please tell me whats wrong with the ...
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2answers
37 views

Is $t\mapsto 1_{[0,t]}(s)$ for a fixed $s\ge 0$ continous?

Let $s\ge 0$ and $$f:[0,\infty)\to\left\{0,1\right\}\;,\;\;\;t\mapsto 1_{[0,t]}(s)$$ Is $f$ continuous at $t_0\ge 0$? If $s>t_0$, then $f(t_0)=0=\displaystyle\lim_{n\to\infty}f(t_n)$ for all ...
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2answers
41 views

under which conditions this equality holds

Consider $f : [0,\infty) \rightarrow \mathbb{R}$ be a function such that $\lim_{t\rightarrow \infty} f(t) = 0$. I was wondering if the following relation holds $$lim_{t\rightarrow\infty}\int_0^t ...
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0answers
25 views

Normal coordinates

I was wondering if this is a legitimate way to define the induced basis of the tangent space in normal coordinates. So the exponential map is a diffemorphism $exp:U \subset T_pM \rightarrow V \subset ...
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28 views

What properties does $a(n)$ have to fullfill to get $\log(2^{cn}+a(n))\sim\log(2^{cn})$?

Let $c$ be an exponential growth rate, and $a(n)$ any expression in $n$ (sequence, polynom, function,...). Consider $$ \log(2^{cn}+a(n)). $$ I am asking myself what properties (increasing, ...
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11 views

Polar coordinates: rescale radius and maintain smoothness

I am looking for a transformation that takes a smooth function $f^\in C^\infty(\overline{B}_1(0))$ on the unit ball, say in $\mathbb{R}^2$, and makes it "even smoother" at the boundary. In 1D, ...
0
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0answers
33 views

integral inequalities and continuous functions [on hold]

Let $f$ be a positive, continuous function on $\mathbb{R}$. Let $c\in (0,1/2)$ be a constant and $\lambda>1$. I want to prove that: (1). for any $a\in\mathbb{R}$, there exists $\delta(a)>0$ ...
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1answer
37 views

if $f$ doesn't have any underbound then $\frac1f$ is not Riemann integrable

I am trying to solve this problem: Assume that the function $f \in \mathscr{R}$ where $\mathscr{R}$ denotes the set of all functions that are Riemann integrable over $[a,b]$.And assume that for some ...
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1answer
36 views

Questions on Kolmogorov Zero-One Law Proof in Williams

Here is the proof of the Kolmogorov Zero-One Law and the lemmas used to prove it in Williams' Probability book: Here are my questions: Why exactly are $\mathfrak{K}_{\infty}$ and ...
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1answer
16 views

$\lbrace \lim f_n(x) \rbrace$ is a Borel set if each $f_n$ is borel

Suppose for all $n$ that $f_n:\mathbb{R}\to \mathbb{R}$ is Borel measurable. What follows is an attempt of the proof that $\lbrace x: \lim_{n\to \infty} f_n\rbrace$ is Borel measurable, but I am a bit ...
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1answer
34 views

Wrong derivation of limit of Cesàro mean

It's known that $$\lim_{n\rightarrow\infty}x_{n}=a\Rightarrow\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}x_{i}}{n}=a$$ Consider the following derivation: ...
3
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2answers
64 views

How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$

I need help to prove this in real analysis. I think it uses IMVT, but not sure how to do it. Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that $$ \lim ...
3
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1answer
40 views

Asymptotic ratio of two series

Assume $\{a_n\}$ and $\{b_n\}$ are two positive series such that $$\sum_{n}a_n=\sum_n b_n=1.$$ Assume also for all $n$, $\sum_{k\geq n}a_k\leq \sum_{k\geq n}b_k$ and $$\lim_{n\rightarrow ...
2
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0answers
29 views

Finding area of a spheroid

Let $M=\{(x,y,z)\in \Bbb{R}^3 : (x/a)^2 + (y/b)^2 + (z/c)^2 = 1\}$. Find $\text{vol}_2(M) = \int_M 1 dS$. My attempt: The map $$\Phi:(0,\pi)\times (0,2\pi)\to \Bbb{R}^3\\ \qquad (\varphi, ...
2
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0answers
20 views

showing a sum of $\vert f(x+k)\vert $ belongs to $L^{\infty}$ if $f,f'\in L^1$

I am workong on this Suppose that $f,f'\in L^1(\mathbb{R})$. Then $\sum_{k= 0} ^{\infty}\vert f(x+k)\vert\in L^{\infty}([a,b])$ for any $a,b\in \mathbb{R}.$ Idea: Let $i$ be any integer. $\int_i ...
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0answers
32 views

Holomorphic function in unit disc?

I am stuck on this exercise from Stein and Shakarchi's Real Analysis: suppose $F$ is holomorphic in the unit disc, and $$\sup_{0\leq r < 1} \frac{1}{2\pi} \int_{-\pi}^\pi \log^+ ...
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2answers
50 views

Finding $n$th root of 2 is irrational using given polynomial

The polynomial $f(x)$ is defined by $f(x)=x^n + a_{n-1}x^{n-1}+ \cdots + a_{2}x^2+a_1x+a_0$ where $n \geq 2$ and the coefficients $a_0, \cdots, a_{n-1}$ are integers, with $a_0 \neq 0$. ...
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0answers
23 views

Harmonic function in $\mathbb{R}^n$ is not one-to-one, for $n\geq 2.$

Le $u:\mathbb{R}^n\to\mathbb{R}$ a harmonic function. Prove that if $n\geq2$ then every $y\in Im\{u\}$ is attained infinite times, but it's not true for $n=1$. I no have idea to start, someone has a ...