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
13 views

Proof of an inequality in spherical harmonics using Minkowski's inequality

I met an inequality in the book "Harmonic analysis and approximation on the unit sphere" by Wang Kunyang and Li Luoqing. I need some hints to follow the proof. The proof is following, \begin{align} ...
0
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3answers
26 views

If $S$ is a subset of a Hilbert space $H$ then $S^\perp$ is closed?

Is this a true statement? (I found it as a theorem in a paper) If $S$ is a subset of a Hilbert space $H$ then $S^\perp$ is closed. If it were true then $(S^\perp)^\perp$ would be closed, that is ...
9
votes
3answers
393 views

Extension of real analytic function to a complex analytic function

I just learned that real analytic functions (by real analytic, I mean functions $f: \mathbb{R} \to \mathbb{R}$ which admit a local Taylor series expansion around any point $p \in \mathbb{R}$) cannot ...
1
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1answer
50 views

finding sup and inf of $\{\frac{n+1}{n}, n\in \mathbb{N}\}$

Please just don't present a proof, see my reasoning below I need to find the sup and inf of this set: $$A = \{\frac{n+1}{n}, n\in \mathbb{N}\} = \{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, ...
-2
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1answer
42 views

How do we show that the sequences is unbounded? [on hold]

how do we prove that if $lim_{n \rightarrow \infty}$ $s_{n}/n = L$, $L \neq 0$ then the sequence $s_n$ is not bounded?
0
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0answers
19 views

Limit Terminology

From the $\epsilon-\delta$ definition of a limit, we can see that any limit can be broken up into two "one-sided limits". These "one-sided" limits are simple cases that arise as a consequence of the ...
6
votes
1answer
53 views

Show $x_n := \frac{1-p^{n+1}}{1-p^n} \frac{n}{n+1}$ is increasing in n for $p \in (0,1)$

I need to show that $x_n := \frac{1-p^{n+1}}{1-p^n} \frac{n}{n+1}$ is increasing in $n$ for $p \in (0,1)$. My attempts have involved trying to show $\begin{eqnarray*} \frac{1-p^{n+2}}{1-p^{n+1}} ...
0
votes
1answer
17 views

Is $\sigma$-finiteness unnecessary for Radon Nikodym theorem?

Let $(X,\mathfrak{M},\mu)$ be a $\sigma$-finite measure space and $\lambda:\mathfrak{M}\rightarrow [0,\infty]$ be a measure. If $\lambda\ll \mu$, then there exists a measurable ...
2
votes
1answer
40 views

Non-usual use of L'Hôpital's rule

I have a very specific doubt about L'Hôpital's rule, from reading a recent paper by Lauermann & Wolinsky in the Jan 2016 Econometrica ("Search with adverse selection".) The authors were ...
2
votes
3answers
45 views

Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find ...
-5
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0answers
27 views

i have a question about real analysis [on hold]

let (X,M,μ) be a measure space. let {E_k} be a countable family of measurable sets such that Σμ(E_k)<∞ prove that μ(X-{x∈X:#{k∈N(natural number) : x∈E_k}<∞})=0 question : is X finite????
0
votes
1answer
46 views

How do I differentiate an improper integral?

I would like to differentiate a function of the type $\int_x^\infty f(x, t) dt$ with respect to $x$ ($f$ real or complex valued). Does differentiation under the integral sign apply? What are better ...
0
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0answers
25 views

Does Cauchy's Multiplication Theorem Apply to Formal Power Series?

Suppose we have two formal power series: $$X(z) = \sum_{n=1}^{\infty} x_n{z_1}^n$$ $$Y(z) = \sum_{n=1}^{\infty} y_n{z_2}^n$$ Then $$\mathrm{exp}(X(z)) = \sum_{n=0}^{\infty} \frac{B_n(x_1,x_2, ...
-2
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0answers
13 views

Derivation formula in $R^n$ [on hold]

Can someone please give me the formula of $\partial_x^b(x-y)^a$ where $x,y \in R^n$ and $a,b$ are multi_index? Thanks in advance
0
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0answers
20 views

Inequality using Taylor formula

For a polynomial $ V∈R[x_1,x_2,..x_d]$ we consider : $R_V(x)=∑_{1≤|a|≤r}|∂^a_xV(x)|^{\frac{1}{|a|}} $. Using the Taylor formula given by: $V(x_0+t)=∑_{1≤|a|≤r}\frac{∂^a_xV(x_0)}{a!}t^a $ I want ...
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0answers
15 views

Example of oscillatory sequence with sum alternating between $\pm \infty$

I am looking for an example of a sequence $\{a_n\}$ whose sum oscillates between $\pm \infty$. Is it possible to use Alternating series theorem to deduce that this behaviour. The theorem says that if ...
1
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1answer
34 views

Showing this function is $0$ a.e.

I would like to show the following: Suppose that $g \in L^1(\mathbb{R}^n)$ and $\int fg \,d \mu = 0$ for any $f \in C_0(\mathbb{R}^n)$. Then $g = 0$ $\mu$-a.e. I'm stumped on trying to find an ...
0
votes
2answers
61 views

Is there a bijection from $[0,1)$ to $\mathbb R ?$ [duplicate]

I know a continuous bijection from $[0,1)$ to $\mathbb R$ cannot exists but what happens if we lift the restriction of continuous $?$ Can there exists a bijection , not necessarily continuous ...
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0answers
27 views

Necessity of Differential Forms

All the undergraduate and graduate texts on analysis introduce Differential and integral calculus (I will assume this introduction of basic calculus/analysis). Among them, some books also introduce ...
1
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1answer
26 views

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$? [on hold]

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$?
1
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1answer
34 views

Prove $\{f_n(x)\}$ is not continous

$f_n(x) = \left\{ \begin{array}{ll} \frac{1}{n} & \quad x \in \mathbb{Q} \\ 0 & \quad x \notin \mathbb{Q} \end{array} \right.$ Not sure how to show ...
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2answers
39 views

Differentiable manifolds in $\Bbb{R}^n$

Let $S$ be a surface in $\Bbb{R}^4$ which consists of the points which are the solutions of the following equations $x+y+z+t =0$ and $x^2+y^2+z^2+t^2=12$. How can we show that it's a two dimensional ...
1
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1answer
31 views

Understanding dual spaces and Riesz's theorem

Is this a proper statement for the Dual space of a Hilbert space? Let $H$ be a Hilbert space. The set of all continuous bounded linear maps, $\mathcal{L}(H,\mathbb{R})$, is called the dual space. I ...
2
votes
1answer
31 views

Is the unit circle “stretchy” with respect to its norm?

Suppose we have a collection of metric spaces on $\mathbb{R}^n$, each of which has a different p-norm, $1\leq p \leq \infty$. ($p=2$ is Euclidean distance, $p=1$ is taxicab distance, etc.) Then, ...
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0answers
12 views

Can you help me win this Differentiable function on Piecewise function question? [on hold]

The question: For a natural number, $n\geq 2$, define $ f(x) = \begin{cases} \hfill 0\hfill & \text{ if $x\leq 0$} \\ \hfill x^{n} \hfill & \text{ if $x > 0 $} \\ ...
0
votes
1answer
27 views

If $h:\mathbb{R}^2 \to \mathbb{R}$ is some function, and $g(r,\theta)=(r\cos{\theta},r\sin{\theta})$, compute the matrix $Dh_{g(r,\theta)}$.

The question says to express $Dh_{g(r,\theta)}$ only in terms of $r$, $\theta$, and the two entries of $D(h\circ G)_{(r,\theta)}$, but I'm not really sure how. Could someone point me in the right ...
3
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5answers
228 views

Complement of rationals has empty interior

This question refers to How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$ I want to prove that the closure of $\mathbb{Q}$ is $\mathbb{R}$. I am trying to understand the accepted answer, but when ...
1
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2answers
53 views

True or false: There exists a series $\sum_n a_n$ of non-negative terms that is convergent, such that $\sum (a_n)^{5/6} $ diverges.

So far I'm thinking that we can use a p-series, $$\sum \frac{1}{n^p},$$ which converges if and only if $p>1$. How would I then show the series where each term is raised to the power of $5/6$ ...
0
votes
1answer
19 views

Show $\lim\limits_k \int_{A_k} f_k = \lim\limits_k\int_A f_k,\;$ given $f_k \in\mathcal{L}^1(\mathbb{R}^n),\; \lim\limits_k\lambda(A_k\Delta A) = 0$.

Show $\lim\limits_k \int_{A_k} f_k = \lim\limits_k\int_A f_k,\;$ given $f_k \in\mathcal{L}^1(\mathbb{R}^n),\; \lim\limits_k\lambda(A_k\Delta A) = 0$. Here $\{A_k\}$ and $A$ are Lebesgue measurable. ...
2
votes
1answer
43 views

Diffeomorphism from $\mathbb{R}^{2}$ onto $\mathbb{R}^{2}$

I've been dealing with this problem. Show that $$\varphi(x,y)=(x+f(y),y+f(x))$$ is a differmorphism of $\mathbb{R}^{2}$ onto itself when $f:\mathbb{R}\longrightarrow \mathbb{R}$ is a continuously ...
2
votes
1answer
34 views

Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
2
votes
1answer
43 views

Argue that the iterated integral of a continuous function is continuous

Suppose that $f : [a, b] \times [c, d] \to\mathbb R$ is a continuous function. Let $$G(y)= \int_a^b f(x, y) \, dx$$ $$H(x)= \int_a^b f(x, y) \, dy$$ Prove that $G$ is continuous on $[c, d]$ and $H$ ...
1
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2answers
39 views

Preimage of sets, complement of sets, continuity of functions

I just got some simple questions in real analysis regarding preimage and complement of sets and continuity. Suppose $f:X\to Y$, then does $f^{-1} (Y\setminus F)=f^{-1} (Y)\setminus f^{-1} ...
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1answer
16 views

Examples in $W^{1,p}(U)\setminus C(\overline{U})$ and $C(\overline{U})\setminus W^{1,p}(U)$

The following is the trace theorem in Partial Differential Equations by Evans: Let $U$ be a domain (open connected subset) of $\mathbb{R}^n$. Suppose $U$ is bounded and $\partial U$ is $C^1$. Then ...
1
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0answers
42 views

Does this function have a dense graph?

Let $\mathbb Q =\{q_n:n\in\mathbb N\}$ be an enumeration of the rationals. Let $f(x)=\mid\sin(1/x)\mid$ if $x\neq 0$ and $f(0)=0$. Let $g(x)=\sum_{n=1} ^\infty \frac{f(x-q_n)}{2^n}$. Question: ...
0
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0answers
18 views

Puzzling weighted L^2 space

As part of an assignment I'm working with the weighthed $L^2(\mathbb R)$ function space $$L_r^2(\mathbb R):=\left\{f:\mathbb R \rightarrow \mathbb C \; \Big| \; \int_{-\infty}^\infty |f(x)|^2 \, r(x) ...
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0answers
13 views

Approach a length by a BV norm

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $g: \overline{\Omega}\to \mathbb R^+$ defined by $g(x)=f(x)$ if $x\in \Omega$ and $g(x)=h(x)$ if $x\in \partial \Omega$, where ...
2
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0answers
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Prove $\sum_{k=1}^\infty k^{-p}f(kx)$ converges absolutely almost everywhere, where $p>0, f \in \mathcal{L}^1(\mathbb{R})$.

What I've done: $$ \int_\mathbb{R} \sum_{k=1}^\infty k^{-p}|f(kx)| = \sum_{k=1}^\infty \int_\mathbb{R} k^{-p}|f(kx)|dx = \sum_{k=1}^\infty k^{-p}\int_\mathbb{R} k^{-1}|f(y)|dy = ...
0
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0answers
16 views

Lower semi continuity for the norm of the speed

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $\gamma_n: [0,1]\to \overline{\Omega}$ be a sequence of Lipschitz functions which converges uniformly on $[0,1]$ to a Lipshitz ...
1
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1answer
29 views

$C^1$ process and infinite variation process

Let $f\in C^1(\mathcal{R}_+,\mathcal{R})$. Could $f$ be an infinite variation process ?
1
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0answers
33 views

Let $f$ be a real-valued continuous function on $[0,1]$ which is twice continu-ously differentiable on $(0,1)$. Suppose that $f(0) = f(1) = 0$

Let $f$ be a real-valued continuous function on $[0,1]$ which is twice continu-ously differentiable on $(0,1)$. Suppose that $f(0) = f(1) = 0$ and $f$ satisfies the following equation: $$x^2f''(x) + ...
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2answers
36 views

What exactly does f'(x)=0 imply from the definition of differentiability?

Let f be a real valued function satisfying $|f (x) −f (a)| ≤ C|x−a|^γ$, for some γ > 0 and C >0. (a) If γ = 1, show that f is continuous at a; (b) If γ > 1, show that f is ...
0
votes
2answers
31 views

Show that $\bar Y = \bar X$

Let $X$ exists such that it is a subset of the real line, and let $Y$ be a set and $X\subseteq Y\subseteq \bar X$. Show that $\bar Y = \bar X$ My Attempt What I know $\bar{X}$ = adherent points of ...
1
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1answer
35 views

Show that $f_n(x) = \frac {2x+n}{x+n}$ converges uniformly to $f(x) =1$.

Question: Let b ∈ $\mathbb R$ and let $f_n(x): [-b,b]\rightarrow\mathbb R $ be defined by $f_n(x) = \frac {2x+n}{x+n}$ for all n ∈ $\mathbb{N}$. Show that ($f_n$) converges uniformly to $f$ on ...
1
vote
1answer
18 views

RS integral for an almost everywhere function

Given that $f$ is function that is Riemann-Stieltjes integrable and that $f=0$ almost everywhere, how can I show that its Riemann-Stieltjes integral is $0$?
0
votes
2answers
27 views

limits of a function 3

I know these are pretty basic but i could really use your help: $\lim: \lim\limits_{x\to 0} \left(\dfrac{1+x}{ 2+x}\right)^ {(1-\sqrt{x})/ (1-x)}$ $\lim: \lim\limits_{x\to 1}\left ...
0
votes
0answers
8 views

Neumann Poincare operator maps $L^2$ in itself

How can I show that the Neumann-Poincare operator $$ K_{\partial \Omega}[\phi](x) = \int_{\partial \Omega} \dfrac{(x-y) \cdot \nu(y)}{|x-y|^d} \phi(y) \ dy $$ maps $L^2(\partial \Omega)$ in itself (if ...
1
vote
2answers
29 views

Cauchy sequence of natural numbers

Consider the set consisting of all cauchy sequences $a_n$ with $a_n \epsilon \mathbb{N}$ for all $n$. Is the set countable? My idea: It is straight forward to prove that any such cauchy sequence ...
0
votes
0answers
18 views

Inverse function theorem as consequence of Implicit function theorem

I'm using Rudin, and in its proof of the implicit function theorem, it uses the inverse function theorem. I've heard that you can prove the inverse function theorem as a consequence of the implicit ...
2
votes
1answer
42 views

regular Borel measure and regular signed measure

I was working on these two questions for a while, I think they are kind of related but I could not figure out any of them: 1)Let $\mu$ be a regular Borel measure on $[0,1]$ such that ...