For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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Quotient Group Proof Help

Let G and H be finite groups with (|G|,|H|) = 1. If ϕ:G-->H is a homomorphism, show ϕ is the trivial homomorphism. proof: Let G and H be finite groups with (|G|,|H|) = 1. Suppose ϕ:G-->H is a ...
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1answer
34 views

Finding Eigenvalues with Gershgorin-Discs

$A=\begin{pmatrix}-5 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & -5 & 4\end{pmatrix}$ Find the Gerschgorin-discs, where the eigenvalues of $A$ lie. According to the formula if we ...
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2answers
67 views

Showing $a_n=\sin(n)$ does not converge

Show that $a_n=\sin(n)$ does not converge My idea: Take two subsequences: $a_{n_k}=\sin(\frac {\pi k} 2)$ , $a_{n_l}=\sin(\frac {2\pi l} 3)$ So: $\forall n$ : $\lim_{n\to\infty} a_{n_k}=1$, ...
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1answer
27 views

Proof - Fundamental Theorem of Arithmetic using Euclid's Lemma

Let $n \in Z > 1$. Then the expression for $n$ as the product of $\ge 1$ primes is unique, up to the order in which they appear. From Proofwiki. Suppose $n$ has two prime factorizations: ...
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2answers
22 views

Proof of Conjugate Subgroup Isomorphism

Let $G$ be a group, and let $H$ be a subgroup of $G$. Prove that if $a$ is an element of $G$, then the subset $aHa^{-1} = \{g ∈ G | g = aha^-1 \text{ for some } h \in H\}$ is a subgroup of $G$ that is ...
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3answers
36 views

Abstract Algebra subgroup and coset proof help

Let H be a subgroup of G. Show that for any a in G, aH = H if and only if a is in H. proof: Let G be a group and H be a subgroup of G with some a in G. Suppose aH = H. Then e is in H since H is a ...
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1answer
13 views

Prime ideals remain distinct

Let $A$ be a domain (commutative ring with unity and no zero divisors) and let $S$ be a multiplicative subset of $A$. Denote by $S^{-1}A$ the ring of quotients $\frac{a}{s}$ (you define the ring ...
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1answer
107 views

Question about integration on a box

Let $Q \subseteq \mathbb{R}^n$, and $f: Q \to \mathbb{R} $ is integrable over $Q$. $f \geq 0$. if $A \subseteq Q$, then $\int_Q f \geq \int_A f $ Attempt: say $\epsilon > 0$ Let $P_1$ be a ...
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0answers
14 views

Are these solutions to Linear Diophantine Equations too? Where'd they hail from?

(1) Can't the signs - I colored them in red - of x and y be switched? Aren't $x = x_0 - bn/d$ and $y = y_0 + an/d$ also solutions? They satisfy $ax + by = c$? (2) How can I remember these ...
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0answers
14 views

geometry 2 column proof of tangent chord angle corollary

I need to prove 12.23 on this section (http://i.imgur.com/M5iev9K.png) I can use any of the theorems or corollaries before 12.23 but not the ones after it. This is a list ...
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1answer
16 views

Holder's Inequality Proof Verification

Wikipedia outlines a nice proof of Holder's Inequality in the link provided. The fifth sentence in the proof reads: Dividing $f$  and $g$ by $\|f \|_p$ and $\|g\|_q$, respectively, we can assume ...
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0answers
24 views

Proof for uniqueness for ideal multiplication

I am across the following question here: The uniqueness of a special maximal ideal factorization Let R be a domain, and let I be an ideal that is a product of distinct maximal ideals in two ways, ...
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1answer
26 views

How can I show this trigonometric identity?

Using only the basic identities ($\sin^2{A}+\cos^2{A}=1$, $1+\cot^2 A=\csc^2{A}$ and $1+\tan^2 A=\sec^2{A}$) show that: $$ \frac{1}{\csc{A}-\cot{A}}-\frac{1}{\sin A}=\frac{1}{\sin A}-\frac{1}{\csc ...
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1answer
34 views

Proof Validation 2

Suppose lim$_{x\to 0} \frac{\phi(x)}{\psi(x)} = 1$ where $\psi(x) > 0$. Let $\alpha_{mn}$ be a doubly index sequence with property that $\alpha_{mn}$ tends to $0$ uniformly in m as n tends to ...
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1answer
46 views

Spectrum and characters: could anyone please check my proof

I tried to prove the following: Let $A$ be a commutative non-unital complex Banach algebra and $\chi : A \to \mathbb C$ a character. Then $$ \sigma (a) = \{\chi (a) : \chi \in \Omega (A) \} ...
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1answer
72 views
+50

Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]

The unit disk is projected onto the xz-plane, so shouldn’t $x = 1\cos \theta$ and $\color{red}{z = 1 \sin \theta} $? User Semsem below kindly identified the problem: The normal to the disk ...
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3answers
42 views

Proof Validation

Prove the intermediate value theorem using the least upper bound property of real numbers. The statement of intermediate value theorem is as follows: Let $f : [a, b] → R$ be a continuous function, ...
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1answer
61 views

Prove 2-HamiltonianCycle $\in \textbf{NP}$

Just want to verify that I have the right idea here with this hamiltonian cycle question. $HC$ = $\{\langle G \rangle$ | $G=(V,E)$ is an undirected graph such that there is a simple cycle (no vertex ...
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1answer
42 views

Solving $y'-y=2\cos 5t$ using the Laplace Transform

Find the solution to the differential equation, using the Laplace Transform. $y'-y=2\cos 5t$, with initial condition $y(0)=0$. My attempt: First I take the Laplace Transform of each term. ...
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0answers
26 views

Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) dx = 1$ for $t>0$.

Suppose $f \in \mathcal{R}$ on $[0,A]$ for all $A < \infty$, and $f(x) \rightarrow 1$ as $x \rightarrow + \infty$. Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) ...
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3answers
37 views

Calculating $\displaystyle \lim_{x \to 0+} \frac{\log(\cos(x))}{x}$ where the domain of the quotient is $(0, \pi/2)$

Calculating: $$\displaystyle \lim_{x \to 0+} \frac{\log(\cos(x))}{x}$$ where the domain of the quotient is $(0, \pi/2)$ The fist step is setting $f(x)=\log(\cos(x))$ and $g(x)=x$, and verifying ...
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0answers
33 views

Looking for an error in a simple proof

Assume n is an integer. If the square root of n is rational, prove that n is a perfect square. To prove the above statement, I used a trick rather than the standard way of using the unique ...
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4answers
45 views

(Proof Checking) Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$.

Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$. I am tempted to use the following argument. Let $U = \{U_i|i\in I\}$ be some open cover of $C\cap ...
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3answers
19 views

Shouldn't the yellow marked $a_0$ be $a_0+\langle p(x)\rangle?$

I'm having problem in getting the proof from Gallian text in the higlighted region: Shouldn't the yellow marked $a_0$ be $a_0+\langle p(x)\rangle?$ Edited: Shouldn't the $a_i$'s in the ...
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0answers
37 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
2
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1answer
113 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
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2answers
61 views

Prove by induction $1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n} \ge 1 + \frac {n}{2}$

Prove by induction $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{2^n} \ge 1 + \frac {n}{2}$ I can't explain in words how the left hand side of the equation is achieved soI shall ...
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0answers
29 views

Proving a set is equal to another set

For all sets $A$ and $B,(B-A)=B\cap A^C$. I would like to know if this proof is correct or if I am on the right track. Here it is: Let $b \in B$ such that $b \notin A$ than $b \in B$ and $b \in ...
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2answers
30 views

Help with proof by induction?

Prove $\frac{2(n-c)}{n+1} < 2$ where c is any natural So we assume $\frac{2(n-c)}{n+1} < 2$ is true, and so far I have $\frac{2(n+1-c)}{n+2} = \frac{2n-2c+2}{n+2} = \frac{2(n-c)}{n+2} + ...
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0answers
26 views

geometric construction puzzle is my construction right?

I think I solved my own question of hyperbolic geometry (and circle ) construction problem but am not sure if it is correct, it looks that way but can somebody help me with prooving it? My ...
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0answers
40 views

My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...
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0answers
57 views

My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
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2answers
52 views

$\varepsilon$-$\delta$-definition for continuity of $x^n$

Show that $f:\Bbb R\to\Bbb R,x\mapsto x^n$ with $n\in\Bbb N$ is continuous in $x_0=0$ using the $\varepsilon$-$\delta$-definition. We assume that $$\forall ...
3
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1answer
46 views

The “intersection property” of the symmetric difference metric

$\newcommand{\measure}{\operatorname{measure}}$ The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space: $$d(S,T)=\measure(S\oplus ...
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1answer
21 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
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1answer
30 views

Question about eigenvalues: eigenvalue $f^2 + f = -1 \rightarrow$ eigenvalue $f^3 = 1$

I have to proof: Let $f \in End(V)$. Show that if $f^2+f$ has eigenvalue $-1$ then $f^3$ has eigenvalue $1$. My idea: If $-1$ is the eigenvalue of $f^2+f$ then there exists (per definition) a $v ...
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1answer
41 views

Where exactly is the following process incorrect to yield an impossible answer

I was playing with my calculator and found some strange phenomena. $\cos(\tan(\tan(\tan(\pi/4)))) = 0.75686700166$ Verify here Now when we apply some inverses, then $\tan(\tan(\tan(\pi/4))) = ...
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0answers
11 views

Deducing the rules for inequalities from the order properties.

I'm trying to relearn calculus but going more in-depth this time, meaning doing all exercises and especially the ones dealing with proofs. I'm using the Thomas calculus textbook. Thomas starts by ...
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1answer
61 views

Learning to understand proofs faster?

There are many books, written by highly decorated academics, which feature proofs that I can hardly comprehend in an acceptable amount of time. Roughly each week, it happens that I find myself having ...
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1answer
60 views

Hausdorff distance and intersection

The question is related to the Hausdorff distance between sets, $d_H(S,S')$, which is the greatest of all the distances from a point in one set to the closest point in the other set. Suppose there ...
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0answers
38 views

My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
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0answers
13 views

Bijective operator has dense image

Let $T\colon C[0,1]\to C[0,1]$ be a bijective operator. My (maybe silly) question is if $T$ then has a dense image. I think I can show this in general: Let $f\colon X\to Y$ be bijective. Consider any ...
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2answers
46 views

Using epsilon -delta definition to show that $\lim_{x \rightarrow 2} (x^3 + \sqrt[3]{x}) = 8+ \sqrt[3]{2} $

I'm trying to use the $\varepsilon$-$\delta$ definition to show that $\lim\limits_{x \rightarrow 2} (x^3 + \sqrt[3]{x}) = 8+ \sqrt[3]{2} $. I already know how to prove continuity for cubic root using ...
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3answers
664 views

My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
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0answers
19 views

Continuity of a function on a box $Q$ implies integrability

Let $Q \subseteq \mathbb{R}^n$ be a box, and say $f: Q \to \mathbb{R}$ is continuous, then $f$ is integrable on $Q$. MY ATTEMPT: Since $f$ is continuous on $Q$, then $f$ must be uniformly continuous ...
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0answers
28 views

A question about open “balls”

I've been recently learning Topology and I'm struggling to visualize open balls. For instance, on $\mathbb{R}^2$ and $\mathbb{R}^3$ given a metric like say $d_\infty(x,y)=\sup\{|x_1-y_1|,|x_1-y_2|\}$ ...
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0answers
33 views

Please check these proofs for sets

I would appreciate the insight again for a couple of proofs since I'm learning. These are homework problems in so much as they are problems from the textbook. They are not required by my professor. ...
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0answers
33 views

Convergence of geometric shapes

Let $c$ be a closed geometric shape. Let $P$ and $Q$ be two points in $c$. Let $S$ be the family of all closed squares. Let $s_1, s_2,...$ be a sequence of shapes from family $S$ such that for every ...
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1answer
30 views

$v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)

Hypothesis: Let $u$ and $v$ be two functions from $\mathbb{R}^2$ to $\mathbb{R}$ s.t. $$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$ and $$ \Delta v = ...
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0answers
63 views

Tim Chow's proof that the moser-number is much smaller than grahams number

The link here shows a proof from Tim Chow that the moser-number is much smaller than grahams number. I do not understand the inequality 3^^...^^3 (3^^^^^3×2-1 arrows) << G 2 What does G 2 ...