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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Merely conditionally convergent series

What is the definition of 'Merely conditionally convergent series'? Is it exactly same as 'conditionally convergent series'? Or different??
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A basic O.D.E doubt

Consider the non-autonomous O.D.E $\dot{x}(t) = \int h(x(t),y)\mu(t,dy)=F(x(t),t)$ where $\mu(t,dy) = \delta_{y_n}(dy)$ when $t \in [t_n,t_{n+1})$ where $y_n$ and $t_n$ are given sequences s.t. ...
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3answers
34 views

Limit of $a_n$, where $a_1=-\frac32$ and $3a_{n+1}=2+a_n^3$.

Let $a_1=-\frac32$, and $3a_{n+1}=2+a_n^3$. I need to show that $\displaystyle \lim_{n\to \infty} a_n = 1$. I can show that the sequence is monotonically increasing and bounded as follows: By the ...
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5answers
64 views

If $a,b\in\mathbb R$ with $a<b$, then there is some rational $r$ with $a<r<b$. [duplicate]

How do you prove this question? I was thinking proving contrapositive. But I was stuck..Thanks guys.
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1answer
15 views

singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
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16 views

The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
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94 views

Calculating a limit of integral

Computing the limit: $$\lim_{n\rightarrow\infty}\left(\frac{1}{3\pi}\int_\pi^{2\pi}\frac{x}{\arctan(nx)} \ dx\right)^n$$ I made the substiution $t=nx$ then, we have: ...
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40 views

Limit of a function to the power of another function

Is there a theorem in real analysis for $\underset{n\rightarrow \infty}\lim f(n)^{g(n)}$, where $f(n)$ and $g(n)$ are arbitrary functions of $n$? Under what conditions on $f(n)$ and $g(n)$ does the ...
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27 views

Function continuous at end-points [on hold]

If we have a function $f$ that is absolutely continuous on $(-1,1)$ (and also the derivative of $f$ is absolutely continuous on $(-1,1)$) and we have that for the derivative the following limits ...
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19 views

Question about simple functions as described in Folland's Real Analysis

The statement of the theorem is: If $f=0$ almost everywhere, then $\int f =0$. My question is in the proof in Folland (provided below) it seems that we are using some statement like: If $\phi$ is ...
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27 views

Family bounded in $\mathcal{L}^1$ has limit a.e.

Let $(X, \mathcal{F} , \mu )$ be a measure space. Suppose $\lbrace X_n \rbrace$ is a family of functions in $\mathcal{L}^1$, bounded in $\mathcal{L}^1$ i.e. there exist $K \geq 0 $ such that ...
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78 views

If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous?

Question: If $f(\mathbb{R})$ is compact and $f$ is continuous, then is $f$ uniformly continuous? Background: I thought of the question when proving that "If a function is periodic and continuous, ...
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3answers
120 views

a limit property at infinity

Let $k\in(0,1)$ is fixed and $L$ is a finite value. Is it possible to say if $\lim_{x\to\infty}f(x)=L$ then $\lim_{x\to\infty}f(kx)=L.$
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26 views

Absolute convergence of an infinite series and p-series test

Why does the infinite series $\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}z^n$ where $z\in \mathbb{C}$ converge absolutely for $|z|<1$. Doesn't the series diverge because if we apply the absolute ...
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2answers
58 views

Stone-Weierstrass: Examples

By Stone-Weierstrass one has: $f\in\mathcal{C}(K):\quad p_n\to f$ Now, for analytic functions this is just Taylor: $$f\in\mathcal{C}^\omega([a,b]):\quad f(x)=\sum_{k=0}^\infty a_kx^k$$ But, how does ...
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34 views

Proving least upper bound property implies greatest lower bound property

In Rudin 1.11 Theorem Proof he claims the following Suppose $S$ is an ordered set with the least upper bound property $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of ...
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14 views

The level set of Lipschitz functions

Suppose $u$: $R^N\to R$ is lipschitz, then do we have a.e. level set of $u$ has Lipschitz boundary? Is this anything to do with Sard theorem? Sard theorem states that a.e. Level set of smooth ...
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25 views

Strong law of large numbers for square-integrable and uncorrelated random variables with bounded variance

Let $(\Omega,\mathcal{A},P)$ be a probability space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable and uncorrelated random variables $\Omega\to [0,\infty]$ with ...
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9 views

$L^2$ regularity of a convolution with newtonian potential.

I am reading Bertozzi, Majda Vorticity and incompressible flow and in page 71 72, we are concerned with recovering the velocity field of a flow from its vorticity. At some point we need to have the ...
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32 views

Lebesgue integral over $\mathbb R^2$ of the function $f(x,y)=2(x-y)e^{-(x-y)^2}\chi_{\{x>0\}}$

Let $f:\mathbb R^2\to \mathbb R$ be given by $$f(x,y)=\begin{cases}2(x-y)e^{-(x-y)^2}& \text{ if }x>0 \\0&\text{ otherwise}\end{cases}$$ Given that $\int^\infty_{-\infty} e^{-z^2} dz=\sqrt ...
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3answers
105 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
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72 views

Is $d(x,y)=|x-y|^2$ a distance on $\mathbb{R}$?

Please how to prove that $d(x,y)=|x-y|^2$ is a distance on $\mathbb{R}$, I don't know how to solve the triangular inequality. Thank you.
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1answer
55 views

Proof, that the floor and ceiling functions exist

I want to proof the following, with elementary properties of the integers and reals: Let $x\in \mathbb{R}$. Then there are unique $p,q\in \mathbb{Z}$, such that: $$p\leq x < p+1\text{ and ...
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14 views

Solving this equation: $⌊r^{m²}α⌋-r^{2m-1}⌊r^{(m-1)²}α⌋=d$

Let $r≥4$ be a positive integer. Let $α$ be a positive real number and let $d$ be a positive integer. I want to solve this equation with respect to the variable $m$: ...
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2answers
38 views

Convergence of a integral: $\int_{0}^{1} |\ln (x)|^n \ dx$

Let $n \in \mathbb N$ be arbitrary. Does the integral $$\int_{0}^{1} |\ln (x)|^n \, dx$$ converge? I asked myself this question and I have no idea of a proof or counter example. Someone can give me a ...
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1answer
12 views

On the polar representation of an inner product.

Take $H$ an inner product space. $x,y \in H$. Take $b = |<x,y>|$ . Then the polar representation of $<x,y>$ is: $$<x,y> = be^{i\theta}$$ for some $\theta \in (-\pi, \pi]$. Why is ...
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70 views

Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$

suppose that $\phi(n)$ is Euler function. prove that, $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$ (if $A_n=\{1 \leq m \leq n | m \in \Bbb N ; gcd(n,m)=1\}$ then $\phi(n)=|A_n|$) I ...
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10 views

Multitangent to a polinomial function

I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of ...
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21 views

why the set $A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x-y\in E\}$ is $\mathcal{B}\times\mathcal{B}$-measurable

If $E\in\mathcal{B}$ , then the set $A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x-y\in E\}$ is $\mathcal{B}\times\mathcal{B}$-measurable, where $\mathcal{B}$ is the family of Borel subsets of $\mathbb{R}$ ...
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39 views

Associated Legendre polynomials

The associated Legendre ODE is given by $$ \left( (1-x^2) f'(x) \right)' - \frac{m^2}{1-x^2} f(x) = \lambda f(x)$$ The eigenfunctions have certain properties that I would like to understand by ...
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The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is ...
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1answer
26 views

A basic non-autonomous O.D.E question

Consider the following non-autonomous O.D.E $$ \dot{x}(t) = h(x(t),g(t))$$ such that $h(.,.)$ is continuous but $g(.)$ is discontinuous(step function). Does the solution exist here ? I don't think ...
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37 views

Finite coloring of an interval

Two real functions, $f$ and $g$, are defined on the interval $[-1,1]$. Each point $x$ in the interval is colored in one of 3 colors: Red - if $f(x)>g(x)$ Blue - if $f(x)=g(x)$ Green - if ...
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89 views

Convergence of summable sequences

If $(a_n)$ is a sequence such that $$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$ How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
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69 views

Every sequence in $\mathbb{R}$ has a monotonic subsequence

I have trouble with this kind of infinite construction in topology. Can someone check my proof is sound? Let $s$ be a sequence in $\mathbb{R}$. Then $s$ has a monotonic subsequence. There are two ...
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Properties of the Fourier transform

Suppose $g,h \in L^1(\mathbb{R} / 2 \pi)$ with $g(x)=h(nx)$, $n \in \mathbb{Z}$. I want to show that $$ \widehat{g}(kn)= \widehat{h}(k), \\ \widehat{g}(l)=0, l \not\equiv 0 \ \text{mod} \ n.$$ I ...
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1answer
14 views

Young's inequality implies $L^p$ convergence of convolution

I am reading a material which states: If $f_n \to f$ in $L^1(\mathbb{R})$, $g \in L^P(\mathbb{R})$. Then $f_n*g \to f*g$ in $L^p(\mathbb{R})$ by Young's inequality. But I cannot see why Young's ...
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35 views

Continuous piecewise smooth curve

I cannot understand the definition of $\tilde d(p_1,p_2)$ here? Can anyone please explain it clearly?
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59 views

Finding points that satisfy $f(a) = \sup f(x)$

Choose positive real numbers $\alpha_1,\ldots,\alpha_n$, $n$ such that $\sum_{i=1}^n \alpha_i = 1$ and let $$f: [0,\infty)^n \to \mathbb R$$ $$x=(x_1, \ldots, x_n) \mapsto x_1^{\alpha_1} \cdots ...
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46 views

Application of Rolle's Theorem and differentiation

Suppose $f: \mathbb{R}\rightarrow \mathbb{R}$ is differentiable with $f(0)=f(1)=0$ and $\{x:f'(x)=0\}\subset \{x:f(x)=0\}$. Show that $f(x)=0$ for all $x\in [0,1]$. My Work: By Rolle's Theorem ...
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22 views

Surjective bilinear map

Let $Q$ be a CONVEX quadrilateral in $R^2$ with vertices $a_1,a_2,a_3,a_4 \in R^2$. Consider the bilinear map $f: [0,1]^2 \to Q$ $$f(x,y)=a_1+(a_2−a_1)x+(a_4−a_1)y+(a_1+a_3−a_2−a_4)xy$$ Note that $f$ ...
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Show that there is $f\in L^1(X,\mu)$ with $P(f)<\infty$ and $P(f_n-f)\to 0$ as $n\to\infty$ [on hold]

Could you please help me solving this old prelim problem. Any hints are appreciated
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1answer
45 views

Proving existence of a linear functional

Let $(X, \| \cdot \|)$ be a normed space, and let $A, B ⊂ X$ be disjoint convex sets such that $B$ is closed and $A$ is compact. Prove that there exists $\varphi ∈ X^*$ such that $$\sup_{a\in A} ...
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27 views

Mean value inequality geometrical interpretaion

The mean value inequality theorem Let U be an open interval in $\mathbb{R}$. Suppose that $K \ge 0$ and that, $a,b \in U$ with $b>a$. If $f : U \rightarrow \mathbb{R}$ is differentiable with ...
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84 views

Analysis: Prove divergence of sequence $(n!)^{\frac2n}$

I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $. I've tried different methods but I haven't really got anywhere. Any solutions/hints?
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35 views

Taking derivatives of the implied function - from the implicit function theorem,

I showed that the relation $$f(x,y)=e^x - e^y + xy = 0$$ defines near (0,0) an implicit function y=$\phi (x)$, since the $1x1$ block, $\frac{df}{dy}$, evaluated at (0,0) gives -1, which is non-zero - ...
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42 views

About fractional iterations and improper integrals

Let $g(x,0) = x$ and $g(x,t+1) = g(x,t) - \dfrac{1}{g(x,t)}$ for every real $t$. From the fact \begin{align} ...
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33 views

Two disjoint connected and bounded open sets in the plane that shares the same boundary

In $\mathbb{R}^2$ with std. topology I want to exhibit two open sets that are connected, bounded and disjoint but that have a common boundary. My attempt: Since both my sets need to be bounded, my ...
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18 views

Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
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35 views

Anyone knows a Good Textbook in Numerical PDES

I am planning on taking a course on numerical PDEs next semester. The course covers the following topics listed below. I am looking for a good book that covers these topics (or at least most of them). ...