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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1answer
58 views

$f$ integrable iff $\sum_{n=1}^{\infty} f(n)$ converges absolutely

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
5
votes
4answers
467 views

Ambiguity in definition of compactness

I am struggling with the definition of compactness in a topological sense. Below is the definition presented in my lecture notes: A topological space $X$ is compact if every open cover has a ...
1
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4answers
122 views

Error in solving $\int \sqrt{1 + e^x} dx$ .

I want to solve this integral for $1 + e^x \ge 0$ $$\int \sqrt{1 + e^x} dx$$ I start by parts $$\int \sqrt{1 + e^x} dx = x\sqrt{1 + e^x} - \int x \frac{e^x}{2\sqrt{1 + e^x}} dx $$ Substitute ...
3
votes
4answers
82 views

Limit of $ \frac1x \int_x ^{2x}e^{-t^2}dt$

What is the limit of the function $$\lim_{x\to 0} \ \frac1x \int_x ^{2x}e^{-t^2}dt$$ ? I tried this problem by using gamma function. I couldn't find the integral.
3
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1answer
33 views

Real Analysis, Folland problem 1.4.20 Outer measures

Exercise 20 - Let $\mu^*$ be an outer measure on $X$, $M^*$ the $\sigma$-algebra of $\mu^*$-measurable sets, $\overline{\mu} = \mu^*|M^*$, and $\mu^+$ the outer measure induced by $\overline{\mu}$ ...
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2answers
32 views

Term-by-Term Differentiation and UNIFORM CONVERGENCE: True Relation

For a series with $\sum u_n'(x)$ not uniformly convergent, and If $f '(x) = \lim_{n\to\infty} f_n'(x) $ where $f(x)=\lim_{n\to\infty} f_n(x) $ and $ f_n(x) $ $=u_1+u_2+ . . . +u_n$ Then the ...
2
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2answers
76 views

$f,g$ coincide on $\mathbb N$ iff $f(x)=g(x)$ a.e.

Given a measure space $(\Bbb R, \mathcal P(\Bbb R), \mu_\Bbb N)$ where $\mathcal P(\Bbb R)$ denotes the power set of $\Bbb R$ and $\mu_\Bbb N$ is defined by $\mu_\Bbb N(A)= \vert {A \cap \Bbb ...
7
votes
4answers
379 views

In general, why is the product topology not equal to the box topology

I am trying to understand a counter-example showing that the box topology and product topology are not equal. Here it is: Let $\tau$ and $\tau'$ be the product and box topologies respectively. Let ...
0
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0answers
30 views

Why does Apostol set $v=\lambda y$?

I refer to Apostol's book on Mathematical Analysis, Theorem 12.11. I am puzzled why does Apostol need to set $v=\lambda y$? Is the reason just to ensure that "$\lambda$ is small enough so that ...
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0answers
11 views

Truncation Error of Adams-Bashforth 3 step Method

I'm attempting to derive the truncation error for the 3 step Adams-Bashforth method. I know that to derive the truncation error for the 2 step Adams-Bashforth method we proceed as follows. Suppose ...
5
votes
1answer
60 views

Lebesgue integral - no dominating integrable function of $(f_n)$

Let $\lambda$ be the Lebesgue-measure on $\Omega =[0,1]$. Given a sequence of non-negative measurable functions $$f_n:\Omega\to\Bbb R: x \mapsto ne^{-nx},$$ how can I show that $f_n$ converges ...
0
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2answers
21 views

Pointwise convergence of Lipschitz functions from a compact space implies uniform convergence

Let $(f_n)$ be a sequence of $1$-Lipschitz functions from $(X, d_X)$ to $(Y,d_Y)$ where the first one is compact and the latter is complete (I am not sure if this matters). Let $f_n \to f$ pointwise. ...
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0answers
13 views

Obscure first order approximation

I don't understand this first order approximation from Gelfand, Fomin "Calculus of Variation": $$ \int_{x_0}^{x_0 + \delta x_0} F(x, y + h, y'+h') dx \sim F(x,y,y')\big|_{x = x_0}\delta x_0$$ where ...
1
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1answer
29 views

Construct an isometric isomorphism between $L_p(\mathbb{R})$ and $L_p[0;1]$

I know that $L_p(\mathbb{R})$ and $L_p[0;1], \; p<+\infty$ are isometrically isomorphic, which means that there is an isomorphism that respects norm. The question is how to construct it?
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0answers
27 views

It is possible to prove that the inverse of every periodic function is a multivalued function?

As I state in the title, I wonder if is possibile to prove that the inverse of every periodic function is a multivalued function. First of all I can't found a counterexample for the statement, and ...
0
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1answer
40 views

Prove continuity of piecewise function using epsilon-delta

Suppose we have a function $\phi$ so that $$\phi (x)=\cases{f(x) & \text{ if } x\le 0\\ g(x)& \text{ if } x>0.}$$ where $f$ is continuous on $(-\infty,0]$ and $g$ is continuous on ...
1
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3answers
57 views

changing the order of the logical symbols $(\forall \epsilon) (\exists\delta)$ by $(\exists \delta)(\forall\epsilon) $ in limit definition

Some time ago a professor told the class, which I was in, to analyze why this definition of limit is not good (or if it is a good definition to argument why): There exists a $\delta>0$ for all ...
1
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0answers
50 views

Properties about a function on $R$ [duplicate]

I have no clue about the following (supposedly simple) question from Mathematical Analysis: Let $f: [a,b] \to \mathbb{R}$ be differentiable on $[a,b]$. Suppose that $f'(a)=f'(b)=0$, and that $f''$ ...
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0answers
41 views

Relation between improper Riemann integral and lebesgue integral on $\mathbb{R}^n$

We know Lebesgue integral is not a generalization of improper Riemann integral on $\mathbb{R}$. Now, I need a more general theorem on $\mathbb{R}^n$, satisfying a weak enough condition, then improper ...
2
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0answers
38 views

Help complete this proof on transcendentalism

Proof $\pi*e$ is transcendental. either $\pi + e$ or $\pi*e$ is transcendental to see take $(x-\pi)(x-e)=x^2-(\pi+e)x+\pi*e$. Case 1 assume $\pi$ and $e$ are algebraically independent. It follows ...
3
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1answer
56 views

Limit probability of a complete bipartite random graph $G(n,n,p)$ is connected

I need to calculate the following probability limit for a complete bipartite random graph $G(n,n,p)$ in the Erdos-Renyi model: \begin{equation} \lim_{n\rightarrow\infty}\mathbb{P}[G(n,n,p) \text{ is ...
5
votes
2answers
65 views

n-th derivative where $n$ is a real number?

We know that $$\frac{d^n}{dt^n} e^{at}= a^n e^{at}; \, n\in \mathbb N.$$ I want to know if the result is true if $n$ is a real number, i.e., $n\in \mathbb R$ ?
0
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1answer
49 views

Prove it has a definite solution.

I'm stuck on this problem. I don't even know how to start: If $(f_1,...,f_n) : \mathbb{R}^n\longrightarrow{\mathbb{R}^n}$ with $f \in C^1 $ is a vector field and ...
0
votes
0answers
10 views

Algebraic dependent summation of transcendental numbers.

Question if $a$ and $b$ are both transcendental but algebraically dependent over Q. what do we know about $(a+b)$? In particular is there a way to bring $(xa^y+nb^m)$ where $x,y,n,m$ are rational and ...
2
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1answer
30 views

Proof Using a Countable Union of Nonempty Open Sets to Prove that $\mathbb{Q}$ Is Infinite

I was playing around with an exercise which shows that if $A \neq \varnothing \subseteq \mathbb{R}$ is an open set, then $A \cap \mathbb{Q} \neq \varnothing$. I then extended the problem to a ...
4
votes
5answers
88 views

Proving by induction that $n! < (\frac{n+1}{2})^n$.

As an analysis homework I have to prove by induction that $n! < (\frac{n+1}{2})^n : (2 \le n \in\mathbb{N})$ For $n = 2$ this is trivial, but for $n+1$ no matter how I transform the equation I ...
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1answer
56 views

$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$

How can I prove $$\int\lim_{n\to\infty}f_nd\mu = \lim_{n\to\infty}\int f_n d\mu$$ given a measure space $(\Omega,\mathfrak A, \mu)$, a non-decreasing sequence $(f_n)$ of measurable functions on ...
1
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1answer
41 views

Definition of $\limsup$ from Rudin RCA

Hi! This is from Rudin's RCA about $\limsup$. I would like to check some moments. $1)$ Since $b_k\to \beta$ as $k\to \infty$. For $\varepsilon>0$ $\exists N=N_{\varepsilon}>0$ such that for ...
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votes
2answers
64 views

Unique solution of the equation $ f (x) f (y)=f (p) f (q) $ given $x+y=p+q$ and $ x^2+y^2=p^2+q^2 $

Unique solution of the equation $ f (x) f (y)=f (p) f (q) $ given $x+y=p+q$ and $ |x|^2+|y|^2=|p|^2+|q|^2 $ $\quad$ $f(x) \ge 0$ $ x, y,p, q $ are vectors (Euclidean space) that is each vector has ...
0
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0answers
37 views

Can $f\in L^2(\Omega)$ imply $\nabla f\in [(H^1(\Omega))^*]^n$?

This is closely related to a previous question of mine. The only difference is the definition of $H^{-1}(\Omega).$ Suppose $f\in L^2(G_R)$ where $$ G_R=\{x\in\mathbb{R}^n\mid ...
0
votes
1answer
34 views

Proving an equality using the Leibniz integral rule.

If $f:[0,b]\times [0,b]\to \mathbb{R}$ is continuous, proof $\int_{0}^{b}\mathrm{d}x\left(\int_{0}^{x}f(x,y)\, \mathrm{d} y\right) = \int_{0}^{b}\mathrm{d}y\left(\int_{y}^{b}f(x,y) \,\mathrm{d} ...
0
votes
1answer
42 views

How to show with induction that $A(x) = A'(x)$

If we have the the formal power series $\sum \limits_{k=0}^{\infty} \frac{1}{k!} \cdot x^k$ and his generating function $A(x) = e^x$. How to show with induction, that $A(x) = A'(x)$? First of all I ...
1
vote
1answer
50 views

If $|f_n|\le S_n$ and $\sum_{n=1}^{\infty}f_n$ converges uniformly, does $\sum_{n=1}^{\infty}S_n$ converge?

Let $f_n:D\to\mathbb{R}$ be bounded and nonnegative functions, with $D\subseteq\mathbb{R}$ and $S_n:=\sup\{f_n(x):x\in D\}$ for all $n\in\mathbb{N}$. If $\sum_{n=1}^{\infty}f_n$ converges ...
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1answer
35 views

Domain of function 1.

what is the domain of $$f(x)=\ln(\sin x)$$ for all $x\in\mathbb{R}$ where $\ln$ is the napierian logarithm. Thanks.
2
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2answers
43 views

Convergence and value of improper integral

I have to prove that integral $I = \int_{0}^{+\infty}\sin(t^2)dt$ is convergent. Could you tell me if it's ok? Let $t^2=u$ then $dt=\frac{du}{2\sqrt{u}}$ Now $$I = ...
0
votes
1answer
21 views

Show $supp(T_\mu) = supp(\mu)$ where $T_\mu(\phi) = \int_U \phi d\mu$ for all test functions $\phi$

So first I was able to show that $T_\mu$ is in fact a distribution. To show their supports are equal, I'll look at the complements, and so I need to show that the largest open set on which $T_\mu =0$ ...
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0answers
15 views

Integration by parts and a vector field

The solution to my problem should be just integration by parts, but I can't see it. We have an integral of the form \begin{equation} \int_Uf(y)e^{ikg(y)}dy \end{equation} where $U\subset \mathbb R^n$ ...
3
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0answers
17 views

Do $(1+\varepsilon)$- bilipschitz maps preserve angles?

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a bilipschitz map such that $$ \frac{1}{1+\varepsilon} d(x,y) \leq d(f(x),f(y)) \leq (1+ \varepsilon) d(x,y) $$ for some $\varepsilon>0$ and the ...
0
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0answers
15 views

Lefschetz Fixed Points and Invertible Derivative Matrices

Background: Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be continuously differentiable. Def: $x\in\mathbb{R}^n$ is a fixed point if $f(x)=x$ Def: $x\in\mathbb{R}^n$ is a Lefschetz fixed point if ...
1
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0answers
29 views

Fourier transform without using Lebesgue measure

Let $\mathbb{L}^p(\mu)$ be a space such that $$ \mathbb{L}^p(\mu) = \left\{f:\mathbb{R}\to \mathbb{R} \mbox{ measurable}: \|f\|_{L^p(\mu)} = \left(\int_0^{+\infty} ...
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2answers
43 views

What is the motivation of this inequality?

Problem Let $$S_n = \sum_{k=1}^n \left(\sqrt{1+\frac{k}{n^2}} -1\right)$$ Show that $\lim_{n\rightarrow \infty} S_n = 1/4$. Solution We first observe that for all $x ...
3
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0answers
72 views

How to compute the following integral?

Someone has an idea to calculate the following integral $$I_{a,b,\alpha} = \int_{1}^{+\infty} e^{-at} \,(1-t^{-1})^b \log^{\alpha}(1-t^{-1}) \, dt; \quad a,b>0, -1<\alpha<0.$$ Thank you in ...
0
votes
3answers
33 views

One partial derivative is continuous (at a single point) implies differentiable?

Let $f:\mathbb{R}^2\to\mathbb{R}$ and $(p,q)\in\mathbb{R}^2$ such that both $f_x$ and $f_y$ exists at $(p,q)$. Assume that $f_x$ is continuous at $(p,q)$. How do we prove/disprove that $f$ is ...
0
votes
1answer
25 views

Cauchy sequences are bounded

As $\{x_n\}$ is a Cauchy sequence, there exists a positive integer $N$, such that for any $n \geq N$ and $m \geq N$, $d(x_n,x_m) \lt 1$; that is, $|x_n-x_m| \lt 1$. Put $M = |x_1| + |x_2| + |x_3| + ...
2
votes
0answers
41 views

Condition for term-by-term differentiation of a non-convergent series

In a problem I have seen, a series $\sum_n u_n(x)$ with $$ f_n(x) = \frac {\log(1+n^4x^2)}{2n^2}$$ here $\sum u_n'(x)$ is not uniformly convergent, BUT If $f '(x) = \lim_{n\to\infty} f_n'(x) $ ...
0
votes
0answers
42 views

$f$ is Riemann integrable iff it is continuous $a.e.$

I want to prove every bounded function $f:[a,b]\to \mathbb R$ is riemann integrable iff it is continuous a.e. This is a way that Folland suggests in his book: if $P$ is a partition for $[a, ...
1
vote
2answers
89 views

A function on [a,b] that is second differentiable and f'(a)=f'(b)=0

Let $f:[a,b]\rightarrow\mathbb{R}$ be secondly differentiable and $f'(a)=f'(b)=0$. Then there exits a point $c\in [a,b]$ such that $$|f''(c)|\geq\frac{4}{(b-a)^2}|f(b)-f(a)|.$$ I tried to prove it by ...
0
votes
1answer
45 views

Example of how to use the Intermediate Value Theorem

Theorem: Let $f$ be continuous on $[a,\,b]$ and assume $f(a)\not=f(b)$. Then for every $\lambda$ such that $f(a)<\lambda<f(b)$, there exists a $c\in(a,\,b)$ such that $f(c)=\lambda$. ...
2
votes
1answer
46 views

Real Analysis, Folland problem 1.4.19 Outer Measures

Background information: Exercise 18 - Let $\mathcal{A}\subset P(X)$ be an algebra, $\mathcal{A}_\sigma$ the collection of countable unions of sets in $\mathcal{A}$, and $\mathcal{A}_{\sigma\delta}$ ...
0
votes
0answers
46 views

Does one need a differential equation to do boundary layer theory?

I'm trying to understand permutation theory and in particular the $$z=\frac{x}{\epsilon}$$ substitution to get an inner solution. Here is my toy example: $$f(x)=\sqrt{x} - x^{1/\epsilon}$$ and I'm ...