I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function
I know that to solve it we must first find...
I think the initial assumptions would allow me to prove this without induction.
Suppose ##(x_n)## is a real sequence that is bounded above. Define $$ y_n = \sup\{x_j | j \geq n\}.$$
Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n##
$$ x_{j} \leq y_n.$$...
Summary: Rudin theorem 1.21
He has said that as t=X/(X+1) then t^n<t<1 then maximum value of t is 1. then in the next part he has given that t^n<t<x. as maximum value of t is less than 1 why has he given that t<x ?
Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established:
\begin{equation}
||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),
\end{equation}
with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement:
$$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the...
Homework Statement
We've been given a set of hints to solve the problem below and I'm stuck on one of them
Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0
Hint
let set S = {x∈[a,b]:f(x)≤0}
let c =...
Hi forum.
I'm trying to prove a claim from Mathematical Analysis I - Zorich since some days, but I succeeded only in part.
The complete claim is:
$$\left\{\begin{matrix} f\in\mathcal{C}^{(n)}(-1,1) \\ \sup_{x\in (-1,1)}|f(x)|\leq 1 \\ |f'(0)|>\alpha _n \end{matrix}\right. \Rightarrow \exists...
Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :)
https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5
Homework Statement
2. Relevant equation
Below is the definition of the limit superior
The Attempt at a Solution
I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case.
I know...
Homework Statement
Let us have ##n \geq 3## points in a square whose side length is ##1##. Prove that there exists a graph with these points such that ##G## is connected, and
$$\sum_{\{v_i,v_j\} \in E(G)}{|v_i - v_j|} \leq 10\sqrt{n}$$
Prove also the ##10## in the inequality can't be replaced...
Homework Statement
If I have the two curves
##\phi (t) = ( \cos t , \sin t ) ## with ## t \in [0, 2\pi]##
##\psi(s) = ( \sin 2s , \cos 2s ) ## with ## s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ] ##
My textbook says that they are equivalent because ##\psi(s) = \phi \circ g^{-1}(s) ## where ##...
Homework Statement
Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set.
Homework Equations
n/a
The Attempt at a Solution
Let ##x \in T##, then ##\exists \sigma \in S## such that ##x...
Homework Statement
Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...
I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J...
In Rudin, the derivative of a function ##f: [a,b] \to \mathbb{R}## is defined as:
Let ##f## be defined (and real-valued) on ##[a,b]##. For any ##x \in [a,b]##, form the quotient ##\phi(t) = \frac{f(t) - f(x)}{t-x}\quad (a < t <b, t \neq x)## and define ##f'(x) = \lim_{t \to x} \phi(t)##, if the...
I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...
Homework Statement
Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##.
Homework Equations
N/A
The Attempt at a Solution
Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...
Homework Statement
Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##.
2. Relevant results
Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a...
Homework Statement
Let ##p\in\Bbb{R}##. Then the function ##f:(0,\infty)\rightarrow \Bbb{R}## defined by ##f(x):=x^p##. Then ##f## is continuous.
I need someone to check what I've done so far and I really need help finishing the last part. I am clueless as to how to show continuity for...
Hey,
I tried to construct the derivation of the integral C with respect to Y:
$$ \frac{\partial C}{\partial Y} = ? $$
$$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$
with
$$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z...
Homework Statement
Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero.
2. Relevant definitions and propositions:
3. The attempt at a proof:
Proof:(by construction)
Let...
Homework Statement
"A set ##A\subset [0,1]## is dense in ##[0,1]## iff every open interval that intersects ##[0,1]## contains ##x\in A##. Suppose ##f:[0,1]\rightarrow ℝ## is integrable and ##f(x) = 0,x\in A## with ##A## dense in ##[0,1]##. Show that ##\int_{0}^{1}f(x)dx=0##."
Homework...
This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...
Homework Statement
Question: Let ##I = [0,1]##. Suppose ##f## is a continuous mapping of ##I## into ##I##. Prove that ##f(x) = x## for at least one ##x∈I##.
Homework Equations
Define first(##[A,B]##) = ##A## and second(##[A,B]##) = ##B## where ##[A,B]## is an interval in ##R##.
The Attempt at...
Homework Statement
Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##.
Homework Equations
I've included some relevant information below:
The Attempt at a Solution
So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move...
Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$
I want to solve this weakly formulated pde:
$$
0=\frac{A}{N^{d+1}} \sum_i...
Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##,
$$\frac{\partial}{\partial r_k}\int_V...
Dear friends,
I have found a derivation of the fact that, under the assumptions made in physics on ##\rho## (to which we can give the physical interpretation of charge density) the function defined by
$$V(\mathbf{x},t):=\frac{1}{4\pi\varepsilon_0}\int_{\mathbb{R}^3}...
Can anyone explain the behavior of light I came across as I sat in my lounge this evening having a nice cup of Mocha . Hint ( I am sitting in a room with some led ceiling lights on) can you:
1.Guess how many Led lights I have on
2.Explain the appearance of light which is looking like a typical...