This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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8 views

Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
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2answers
21 views

Alternative proof: show that any metrizable space $X$ is normal - Part 2

This is a follow up to one of my earlier questions I am reading some stuff online and saw a proof as follows Per a comment in Part 1 in linked, We know that $d(C_1,C_2)$ could easily be zero ...
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0answers
27 views

Using Rolle's Theorem to prove that f'(c)=rf(c). [on hold]

Suppose that the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)=0$. Prove that for each real number $r$, there is some $c$ on $(a,b)$ such that $f'(c)=r\cdot f(c)$. ...
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0answers
12 views

proof of preimage of union and intersection of sets

I was learning to proof the following proposition "the inverse image of an intersection or union equals the intersection or union of the inverse image" following these two really good youtube videos: ...
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4answers
444 views

Discrete Math Understanding a proof involving the definition of divisibility

In this first course on discrete mathematics, the instructor provided this following solution to a question. The question was asked us to prove the following (the solution is provided as well): My ...
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1answer
26 views

The intuition behind choosing this length?

In Problem 1, RMO 2004 there is a particular choice of length which leads to the solution, the length being that of the tangent from the foot of the perpendicular to the circle. Just a rundown of ...
2
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1answer
55 views

Question 5, RMO 2003, issue with ratios

In problem 5, RMO 2003 a specific part of the solution depends on the following $$\dfrac{BD}{DC} = \dfrac{AE}{EC} = \dfrac{AF}{FB} = \dfrac{DC}{BD}$$ It is proven that $AB \parallel DE \: , BC \...
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0answers
45 views

Estimate for linear error term

Set $\Omega \subset \mathbb{R}^n$ is open and $f\in C^1(\Omega;\mathbb{R}^n)$. Let $\overline{B}$ be any closed ball in $\Omega$. Since $f'$ is uniformly continuous in $\overline{B}$ there are ...
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0answers
15 views

Linearity of projection of angle

In the book Putnam and Beyond, problem 252 reads as follows: Consider the angle formed by two half-lines in three-dimensional space. Prove that the average of the measure of the projection of the ...
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3answers
78 views

I don't understand how this proof works.

My book proves "if $a(x)$ has degree $n$, it has at most $n$ roots." I will just copy the proof here. Proof: If $a(x)$ had $n+1$ roots $c_1, \dots,c_{n+1}$, then by Theorem 2, $(x-c_1) \dots (x-c_{n+...
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1answer
35 views

Discrete Math Proof verification: products of floor

Determine if the following is true or false and provide a proof: $\forall x\in\mathbb{R},\exists y\in\mathbb{R}$ so that $\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor$ My attempt: -The ...
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1answer
30 views

Class equation and orbit stabilizer theorem

I was reading the proof of the following theorem but I cannot understand how to use the class equation as he wants me to. Theorem Suppose that $G=HK$ where $H$ is a normal locally finite $p'$-...
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3answers
66 views

Law of total expectation?

Apparently $E[X] = E[E[X\mid Y]]$ but I don't understand what this really means. I looked at https://en.wikipedia.org/wiki/Law_of_total_expectation but need another explanation. Isn't this the same ...
4
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1answer
57 views

Proving $\text{Var}(X) = E[X^2] - (E[X])^2$

I want to understand something about the derivation of $\text{Var}(X) = E[X^2] - (E[X])^2$ Variance is defined as the expected squared difference between a random variable and the mean (expected ...
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2answers
45 views

An unexplained condition on $a$ in a proof on the primes?

Lemma A positive integer $n$ is a prime if $(n,p) = 1$ for every prime integer $p \leq \sqrt{n}$ Proof in my text Let $(n,p) = 1$ for every prime $p \leq \sqrt{n} \:$. Suppose $n$ is not a ...
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1answer
30 views

Why is $<T\vec x,\vec y>=<\vec x,T^*\vec y>$ for hermitian matrix T?

In this video in 5:15 there's a proof that every hermitian matrix has real eigenvalues. I don't understand the step: $<\vec x, L\vec y>=<L^*\vec x,\vec y>$. I know that I can pull out ...
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0answers
26 views

Proof for $|z_1+z_2| \le |z_1|+|z_2|$ and $|z_1-z_2|\ge |z_1|-|z_2|$ [closed]

I need proof for $$|z_1+z_2| \le |z_1|+|z_2|$$ and $$|z_1-z_2|\ge |z_1|-|z_2|$$
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1answer
24 views

How to prove this using laws? [closed]

How do they prove this? $$(p\to q)\land[\neg q\land(r\lor\neg q)]\equiv\neg q\land\neg p$$
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1answer
65 views

Need help with proof about Diophantine equations

The way I am planning to arrange this is by providing fragments of the proof, so I can understand what's going on before forging ahead, so if you are going to help me, keep in mind that I am going to ...
1
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1answer
16 views

Understanding a projection function from S to B.

Let A and B be sets and $S \subseteq A \times B $ . Let $\pi_{1}$ be the projection function on $S$ to $A$ and $\pi_{2}$ be the projection function on $S$ to $B$. Give an example to show that. $\...
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1answer
34 views

How we can show $\mathbb{E}[T]=0$ and $\operatorname{Var}(T)=\frac{n}{n-2}$.

I need help with this question. Let $Z\sim N(0,1)$ and $Y\sim X^2_{(n)}$ be independent variables, and define$$T\stackrel{\rm def}{=}\frac{Z}{\sqrt{\frac{Y}{n}}}.$$ Prove that $\mathbb{E}[T]=0$...
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2answers
76 views

Proving that $\lceil f(x) \rceil$ $=$ $\lceil f(\lceil x \rceil )\rceil$ when $f(x) =$ integer $\implies x =$ integer

On P. 71 in 'Concrete Mathematics' the following Theorem is given: Let $f$ be any continuous, monotonically increasing function on an interval of the real numbers, with the property that \begin{...
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0answers
18 views

Transpose of the reduced row echelon form

I am reading the proof of a theorem and there is the following statement: Let $U$ be the reduced row echelon form of $A$ and let $V$ be the reduced row echelon form of $U^T$. Then $V^T$ is also ...
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1answer
44 views

How do you prove that any two integers will always have a greatest common divisor?

I think my book actually prove it by showing that the set of all linear combinations of two integers is a principal ideal of integers. But I can't see how this proves that two integers will always ...
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1answer
11 views

achievability of average

Out of a textbook. Informally, the goal is to show that if from a given set of values ($2^n$ many values) in the range of $[0,m]$ (for fixed $m,k$), more than half are less than $m\cdot (1-\frac{1}{2^...
6
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1answer
565 views

Disproving existence statements

I am trying to get some practice on disproving existence statements and I was really stuck on this one: "There exists an example of three distinct positive integers different from $a,2a,3a$ for some ...
2
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1answer
28 views

Question about the proof of the fact that minimum is attained for a l.s.c. convex function over a convex compact set.

I quote here the proof of a result given in Haim Brezis Functional Analysis, Sobolev Spaces and partial differential Equations: I haven't been able to conclude where exactly is this hypothesis used: ...
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1answer
24 views

Convergence of composition of functions in $L^p$

I am dealing with the proof of proposition $9.5$ given in Haim Brezis' Functional analysis, Sobolev Spaces and Partial differential equations. I quote it here: How does one conclude $G \circ ...
2
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1answer
59 views

When do differential operators commute?

Given that the equation of motion of a particle placed on the apex of Norton's Dome is $$\frac{d^2 r}{dt^2}=r^{1/2}\qquad\longleftarrow\text{as proved in this previous question}\tag{1}$$ Solve this ...
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1answer
228 views

Arnold's proof of Abel's theorem

I'm seeking help understanding this video. The author considers the equation $ax^5+bx^4+cx^3+dx^2+ex+f = 0$ and shows both the coefficients $a, b$... and solutions $x_1, x_2$... in the complex ...
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2answers
49 views

Show that the equation of motion for a particle on Norton's Dome is $\frac{d^2 r}{dt^2}=r^{1/2}$

A particle sits at the top of a dome, whose height drops away from the centre, with a drop given by $$h=\frac{2r^{3/2}}{3g}$$ where $g$ is the acceleration due to gravity, and $r$ is a coordinate ...
2
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1answer
36 views

$a_i \mid r $ implies that $r = 0$ if $0 \leq r < a$?

If $x$ is any common multiple of $a_1, a_2 \cdots a_n$ all $\neq 0$ then prove that $[a_1, a_2,\ldots,a_n]$ divides $x$. Note, $[a_1, a_2,\ldots,a_n]$ is LCM. The solution provided in my text: Let $...
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1answer
25 views

Question about the proof $X'$ reflexive $\Rightarrow X$ reflexive.

I have a doubt in the proof I have been given of the fact: For a Banach space $X$, if $X'$ is reflexive then $X$ is reflexive. This is proven by showing first theorem 1 and theorem 2, which I quote ...
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3answers
74 views

Proving the Fermat's theorem

Fermat's theorem: if a is not divisible by p, then $a^{p-1} \equiv 1 \pmod p$ Since $\varphi(p)=p-1$, this is a special case of Euler's theorem. If $(a,m)=1$, then $a^{\varphi(m)}\equiv 1 \pmod m$. ...
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1answer
33 views

Question about Brouwer degree under uniform convergence.

I was wondering the following: Say a smooth sequence $u_k$ on a smooth manifold converges uniformly to the limit $u$. Does $u$ preserve the Brouwer degree of the $u_k$'s? I also believe this is an ...
3
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1answer
33 views

Not understanding the proof that there is no surjection from a set to its powerset

Here is the question: If a set, $A$, is finite, then $|A| < 2^{|A|} = |P(A)|$, and so there is no surjection from set $A$ to its powerset. Show that this is still true if $A$ is infinite. ...
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0answers
52 views

Movie math, real or not? (Ghost busters trailer) [closed]

I was watching the ghost busters trailer, and I stopped on a frame containing 'quantum physics'. I wonder if all of this is meaningful math, or some of it is complete gibberish, to look smart. I see ...
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1answer
20 views

Calculating eigenvalues of an infinite sequence

Consider the set V of all infinite sequences in $\Bbb{F}$. $$V = { \{ (a_0, a_1, a_2, ...)\ |\ a_i \in \Bbb{F}, \text{for } i=0, 1, 2, ... \} }$$ and the function T:V $\rightarrow$ V defined by $$T(...
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2answers
25 views
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1answer
22 views

How can you prove a quadrilateral given a diagonal and segments going from the vertices to the diagonal (picture)?

I'm having quite a bit of trouble with this proof. The angles formed by the segments between diagonals and vertices are 90 degrees, and the vertices on the diagonal and the diagonals' vertices are ...
0
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2answers
20 views

Why the set of points satisfying (2x-x²-y²)*(x²+y²-x)>0 is the same of the set with condition ((2x-x²-y²)>o and (x²+y²-x)>0)?

Why the set of points on $\Bbb R^{2}$ satisfying $(2x-x²-y²)(x²+y²-x)>0$ is the same of the set of all elements on $\Bbb R^{2}$ with condition $((2x-x²-y²)>0 $ and $(x²+y²-x)>0)$, i think ...
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Formal proof on continuity of multivariable function.

Let $B(o,r) = ${$(x,y)\in \Bbb R: \left\lVert (x,y) \right\rVert<r$}$ $ , to some r>0 and the norm is an euclidean norm, let $f(x,y)$ -> $L$ as (x,y) -> (0,0), with f : $B(o,r)$ -> $\Bbb R$. ...
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0answers
47 views

Intuition behind the proof of the validity of the Euclidean algorithm

As the question title suggests, could anybody explain to me their intuition behind the proof of the validity of the Euclidean algorithm?
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1answer
57 views

Need help in understanding some basic ordinal concepts

I'm trying to construct a proof that there is no homeomorphism from a subspace of $\omega_1$ to $\Bbb{Q}$ with the usual topology, but first I need to clear up some concepts. This is the definition ...
0
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1answer
24 views

Sequence of closed sets (Milnor's proof)

I've got a question about a descending sequence of closed sets. Milnor writes in his book "FROM THE DIFFERENTIABLE VIEWPOINT". In his proof of Sards Theorem he wrote: Let $f:U\rightarrow \mathbb{R}^p$ ...
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5answers
44 views

Proof by Mathematical Induction for all natural numbers n.

$1^3 + 2^3 + \cdot \cdot \cdot+ n^3 = $ $[ \frac{n(n+1)}{2}]^{2} $ $\text{My question for this problem is that I got stuck at a certain point}$ $\text{and I do not know where to go. This is what I ...
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2answers
44 views

I don't understand a little section of the Fundamental theorem of arithmetic.

I understand the theorem from a general point of view, but there's this little part, which I don't. This is the theorem, as explained by Richard Courant and Herbert Robbins: What I don't ...
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1answer
28 views

Theorem on arithmetic of natural numbers.

From "Analysis I"-Herbert & Joachim: (starting from the Peano axioms) "There are operations addition + , multiplication · and a partial order ≤ on N which are uniquely determined by the ...
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1answer
30 views

Putnam and beyond section 2.3.2 example

In section 2.3.2 of the book "Putnam and Beyond" there is an example problem attributed to D. Andrica, for which I think the provided proof has an error. In particular, it is asserted (on pg. 65) ...
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0answers
27 views

Definition of hyperbolic lenght.

Theorem 1: Let $\text{arc(AB)}$ be an arc of an equidistant curve (Which can be a circle, a horocircle or an equidistant line) and $(A^{n})$ a sequence of partitions of the arc $\text{arc(AB)}$ such ...