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

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How to prove that $\gcd(a,m) \le gcd(a,mn)$ for any integer n

I'm trying to show that $\gcd(a,m) \le gcd(a,mn)$ for any integer n Taking a classical algebra course and can not seem to figure out how to prove this. I know about Bezout's Identity but don't know ...
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1answer
12 views

Independence Number Proof Explanation

In the following proof it states that "$v_i$ is less than or equal to the independence number for all $i$." Why is this true? I know what an independence number represents, I am struggling to ...
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1answer
15 views

how to prove the following propositional formula using semantic equivalence

Hi Guys I am trying to prove the following formula using the rule below ...
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1answer
37 views

Proof that every non-empty subset of a woset (X, $\leq$) has a unique minimal element.

I want to prove that every nonempty subset of a woset (X, $\leq$) has a unique minimal element. What I’m looking for: clarification and/or hints. I want to solve it on my own, but this is all the ...
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1answer
22 views

Under what conditions can we move the limit symbol through the logarithm symbol?

I was reading the derivation of the derivation of a log function. And saw this: $$\frac{d}{dx}[\log_b x]= \frac{1}{x}\lim_{v \to 0} [\log_b(1+v)^\frac{1}{v}]$$ Then, the limit notation gets moved ...
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Proof: For every positive integer $n$, there is a sequence of $n$ consecutive positive integers containing no primes. [duplicate]

Let $x=(n+1)!+2$. I get how to prove that $x$ or $x+1$ is prime, but there is a step in my book that proves that $x+i$ is prime like this: $x+i=(1)(2)(3)(4)....(n+1)+(i+2)$. But then it factors out ...
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2answers
50 views

Is There a Problem with This Professor's Proof Concerning Interior and Boundary Points?

Here is a professor's solution to the exercise which states, " Prove that if $x$ is an isolated point of a set $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$." The professor derived ...
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2answers
29 views

$U(n)^2$ is a proper subgroup of $U(n)$

I'm trying to show that $U(n)^2$ is a proper subgroup of $U(n)$. Here $$ U(n)^2 = \{x^2 \mid x \in U(n)\}$$ where $U(n)$ is the group of units modulo $n$. My idea was to argue as follows: Consider ...
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4answers
58 views

Misconception of infinite prime numbers proof by contradiction?

I'm using the proof on this page, except with $q$ instead of $p$ on the left side. The misconception of the proof is that $q$ has to be a prime number. I found this using $n = 6$, which gets me $q = 1 ...
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1answer
31 views

Proving that a group representation is *not* a direct sum of irreducible represenations.

Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow ...
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1answer
24 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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1answer
40 views

Show a function is $K$-Lipschitz

The following is the proof taken from Lemma $13$. Questions: $(1)$: What is the Lipschitz-norm of $\phi_a$? The following is my attempt: $\| \phi_a \|_{Lip} = \sup_{x \neq y}{\frac{|f(a) - ...
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0answers
25 views

Lang's proof of the Weierstrass preparation theorem

Relevant Google Books link. I'm having problems with the final step in the proof of Theorem 9.1. It's not clear to me why the function $I + \tau \circ \frac{\alpha(f)}{\tau(f)}$ should be ...
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1answer
48 views

Producing a $G$-invariant form from the standard Hermitian product using the averaging process

Problem statement: Let $G$ be a cyclic group of order $3$. The matrix $$A=\begin{bmatrix} -1 && -1 \\ 1 && 0 \end{bmatrix}$$ has order $3$, so it defines a matrix representation of ...
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2answers
57 views

How do I show that $n=2$ is the only integer satisfy :$\cos^n\theta+ \sin^n\theta=1$ for all $\theta$ real or complex?

It is well known that :$\cos²\theta+ \sin²\theta=1$ for all $\theta$ real or complex ,I would like to ask about the general equality :$\cos^n\theta+ \sin^n\theta=1$ if there is others values of the ...
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1answer
46 views

Help understanding theorem proof

So this is my first semester taking a Real Analysis class. We are using the book Introduction to Analysis by Gaughan 5th ed. This is my first real Math class and I'm really excited but I am having ...
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2answers
28 views

Help Proving the Average is greater than B^(1/n)

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers: $A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers) $B = (a_1 + a_2 + \cdots + ...
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0answers
28 views

Question on a proof involving tightness and almost sure convergence of a sequence

I'm having a hard time understanding the proof of Lemma 17 in this article. Essentially, the assertion of the lemma boils down to replacing a constant in a sequence of random variables that satisfies ...
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2answers
25 views

Universal Instantiation and Proof of: if $x$ and $y$ are odd integers, then $xy$ is odd.

I have a question about the first part of the proof for the statement "If $x$ and $y$ are odd integers, then $xy$ is odd" if we are using a direct proof. Now i've gotten the proof roughly correct but ...
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2answers
50 views

Why is $x^2 \equiv 1 \pmod{x+1}$ for $x > 0$?

One day my mind wandered off and came upon the following. $x^2 \equiv 1 \pmod{x+1}~\forall x>0, x \in \mathbb{Z}$. My markdown might be a little bit broken :) I tested this out in Python for the ...
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1answer
29 views

The set $W^{⊥⊥}$ in a Hermitian space

Problem Statement: Let $W$ be a subspace of a Hermitian space $V$. Prove that $W^{⊥⊥}=W$ I am trying to figure out a good strategy for this proof. I know that: $W$ is a subspace of $V$ ...
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1answer
12 views

Why is the inf-convolution of lower semicontinuous functions continuous?

I'm confusing now about the continuity of inf-convolution. I understand that the inf-convolution of lower semicontinuous functions is semiconcave and so it's locally Lipschitz continuous (in ...
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1answer
42 views

What is the simplest way to show that ${(p-1)! \over (k)!(p-k)!}$ is an integer?

In the proof of $p$ | $\binom{p}{k}$ (p divides $\binom{p}{k}$) where $p$ is prime, what is the simplest way to show that $${(p-1)! \over (k)!(p-k)!}$$ is an integer?
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15 views

Prove that a function is continuous (square integrability)

I need help for the following proof of continuity: Let $E=L_2([t_0,t_1],\mathbb R)$ be a Hilbert space of square-integrable real-valued functions on $[t_0,t_1]$. Let ...
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0answers
13 views

proof for the equivariance of the MLE

I am self-learning statistics and I read about the theorem that under some conditions the MLE is equivariance. I couldn't find any proof for that theorem. What are the conditions and what is the ...
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1answer
41 views

To show a map is an isomorphism

In the proof of Lemma $15$, the author claimed that if there exists constants $C>0$ and $D>0$ such that $$C \sup \{ | \sum_{i=1}^k{\alpha(i) f(x_i) | : \| f \|_{\infty}^1 \leq 1, f(0)=0} \} ...
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1answer
17 views

Inductive factorial formula proof - can't figure out how to finish proof

I am reading The Algorithm Design Manual and in the induction section of the first chapter, I am struggling to figure out how you go from one line to another in the final proof. Here is a picture of ...
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1answer
62 views

Explanation of a proof about graded module structure

Let $\Bbb F$ be a field and $M$ a finitely generated $\Bbb F[x]$-module. The structure theorem for modules over a PID says that $$ M\cong \Bbb F[x]^r\oplus\biggl(\bigoplus_{j=0}^s\Bbb ...
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1answer
26 views

Need a hint for a proof using the pumping lemma that a language is not regular

Currently I am stuck at a proof of : $A_2=\{w001;|w|_0<|w|_1 \wedge w\in \sum^*\}$ unsing only the pumping lemma. Can you give me a hint for a good start? thanks in advance.
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38 views

'Multiplying' by 0 in a field, field axiom proofs

The question says: The solution set was posted and there are a few things I don't quite understand from it. For the first one, I'm not entirely sure what's happening. It appears to be using the ...
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1answer
45 views

Borel lemma : wikipedia proof

In the proof of Borel's lemma, I don't understand why we use $\psi\left(\frac{t}{\epsilon_m}\right)$ for a sufficient small $\epsilon_m$ and not $\psi(t\cdot \epsilon_m)$, as you need to keep ...
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1answer
60 views

Can one use the Hilbert-Ackermann Consistency Theorem to prove the consistency of $PRA$?

In his textbook Mathematical Logic, Shoenfield states the Hilbert-Ackermann Consistency Theorem as follows: "Consistency Theorem (Hilbert-Ackermann): An open theory $T$ is inconsistent iff there is ...
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1answer
32 views

Separation of Independent Probabilities with Condition

This question comes out of the following proof. Two independent random variables, $X_1, X_2$. $X_1$ occurs with probability $p$ and $X_2$ with $(1-p)$. The sum, $N = X_2+X_1$ has $P(N=i+j) \sim ...
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20 views

Hamiltonian Ore Property Proof Clarification

If G has order $n \ge 3$ and for all pairs of distinct vertices $x$ and $y$ that are not adjacent, $deg(x)+deg(y) \ge n$ then the graph is Hamiltonian. Here is an image of the proof- I'm hoping for ...
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1answer
14 views

Hamiltonian Ore Property Proof, why must be connected?

If G has order $n \ge 3$ and for all pairs of distinct vertices $x$ and $y$ that are not adjacent, $deg(x)+deg(y) \ge n$ then the graph is Hamiltonian. Here is the beginning of the proof: We ...
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15 views

Given $f:[a,b] \rightarrow R$, if $x'$ is a local minimum of $f$ and $x'<b$ then there exists a sequence $x_n$ converging to $x'$ with $x'<x_n<b$

I'm trying to understand the demostration of the folowing lemma: Is a function $f:[a,b] \rightarrow R$ differentiable in a local extrema $x'$ then $f'(x')=0$ Demostration: $x'$ is a local ...
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1answer
32 views

What is the min-max argument in mathematics?

In the proof of a theorem the author says that he would prove a special case using the min-max argument. After reading the proof I could not infer what the min-max argument actually does. Could ...
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1answer
87 views

Matrix induction proof

Given the following $\lambda_{1}=\frac{1-\sqrt{5}}{2}$ and $\lambda_{2}=\frac{1+\sqrt{5}}{2}$ How do I prove this using induction: $\begin{align*} A^k=\frac{1}{\sqrt{5}}\left(\begin{array}{cc} ...
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2answers
46 views

Proving double inequality by cases

I am puzzled by an exercise in Vellemen's How To Prove It, here is the exercise: Prove that for all real numbers $a$ and $b$, $|a| \le b$ iff $-b \le a \le b $. There is actually no official ...
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1answer
24 views

Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?

When I read Lee's Riemannian Manifolds : An Introduction to Curvature, I am confused by the red line in the picture below. Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?
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1answer
26 views

Free group action on $S^n$ proof in Hatcher

Theorem: :$\mathbb{Z}_2$ is the only nontrivial group that can act freely on $S^n$ if $n$ is even. Proof: Since the degree of a homeomorphism must be $\pm 1$, an action of a group on $S^n$ determines ...
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Any proper face is contained in some facet.

Definition: A convex polyhedral cone is a set $$ \sigma = \{r_1 v_1 + \dots + r_s v_s \in V \mid r_i \ge 0\}$$ A face $\tau$ of $\sigma$ is the intersection of $\sigma$ with any supporting ...
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0answers
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Sphere inside cylinder vs polyhedra?

Comparing a cylinder with a polyhedra that has a symmetric coxeter $\ge 3$. Both have their centers hollowed out by $k\%$, in the shape of their outer, i.e.: relative to top face Which can better ...
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1answer
31 views

Understanding a proof about nested nonempty connected compact subsets

I know this question has asked to death here on MSE but I have not found a satisfactory solution. A solution found online is extremely elegant but I do not quite understand it! Given nested ...
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2answers
28 views

Proving two integers of opposite parity have an even product?

I think I might be beginning to wrap my head around some simpler proofs, but I'm a little stumped on this one from my textbook: Use a direct proof to show that if two integers have opposite ...
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3answers
51 views

Proving that if $a|b$ and $b|a$, then $a = \pm b$ for $a,b$ as nonzero integers?

Can someone walk me through how to do a proof of the following? Let $a$ and $b$ be nonzero integers. Use a direct proof to show that if $a|b$ and $b|a$, then $a= \pm b$. So I know $a,b \neq 0$ in ...
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1answer
43 views

For all sets A and B, if A ⊆ B, then A ∪ B ⊆ B

I am trying to solve a proof, but I'm a little lost on how to structure it. I have the following setup, but I'm not sure what to put in most of the blank spaces. Proposition: For all sets A and B, ...
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40 views

Using a Direct Proof to show that two integers of same parity have an even sum?

I seem to be having a lot of difficulty with proofs and wondered if someone can walk me through this. The question out of my textbook states: Use a direct proof to show that if two integers have ...
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1answer
33 views

Determining which elements of a matrix group form a one-parameter subgroup

We have just learned about one-parameter subgroups in my Algebra class and I am not sure if I am approaching the following proof in the right way. Problem Statement: Let $G$ be a group of real ...
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1answer
28 views

Show $\left(\vec{A}\cdot\nabla\right)\vec{A} = \nabla\left( \frac{A^2}{2} \right) - \vec{A}\times\left(\nabla\times\vec{A}\right)$

\begin{equation} \begin{aligned} \left(\vec{A}\cdot\nabla\right)\vec{A} &= \nabla\left( \frac{A^2}{2} \right) - \vec{A}\times\left(\nabla\times\vec{A}\right)\\ ...