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Questions tagged [proof-explanation]

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

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If H is a subset of G, prove that H is also a subgroup.

$G$ is a group, and $H$ is a nonempty subset of $G$. We know that, for any two values $a$ and $b$ in $H$, $ab^{-1}$ is also in $H$. Given this, how do we know that $H$ is also a subgroup of $G$? More ...
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Proof of Lemma: Every nonzero integer can be written as a product of primes

I'm new to number theory. This might be kind of a silly question, so I'm sorry if it is. I encountered the classic lemma about every nonzero integer being the product of primes in a textbook about ...
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0answers
15 views

Big rearrangement theorem/Big associative law

Translated from German Großes Assoziativgesetz/Grpßer Umordnungssatz. I am currently reading the textbook analysis 1 from Konrad Königsberger. It is page 70 I have Problems with. I don't know the ...
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Why is one optimal value greater than or equal to the other one here?

Let the first program be $$\min \frac{c^Tx + d}{e^Tx+f} \text{ subject to }\{Gx \le h, Ax = b\}$$ It can be transformed to equivalent linear program: $$\min (c^Ty + dz) \text{ subject to } \...
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2answers
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A variational inequality satisfied in a Hilbert space

I'm trying to establish the existance of $u \in K \subset V$, a closed convex subset of vector space with an inner product, such that for a fixed $q \in V$: $$(u-q,v-u) \geq 0 \quad \forall v \in K$$ ...
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INDUCTION why is this a valid proof?

I have a problem to understand k in the following induction proof. Prove that $3^n -1$ is even for any natural number $n$. Is there anybody that can show me that this is a valid proof and what is ...
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3answers
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Reaching upon $9=1$ while solving $x$ for $3\tan{(x-15^{\circ})}=\tan{(x+15^{\circ})}$

$x$ for $3\tan{(x-15^{\circ})}=\tan{(x+15^{\circ})}$ Substituting $y=x+45^{\circ}$, we get $$3\tan{(y-60^{\circ})}=\tan{(y-30^{\circ})}$$ $$3\frac{\tan y - \sqrt3}{1+\sqrt3\tan y}=\frac{\tan y - ...
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Proof of an identity for the Killing form involving derivations.

I'm working through Ziller's Lie Groups. Representation Theory and Symmetric Spaces, and in Proposition 1.36, he shows the following identity: Let $\mathfrak{g}$ be a real or complex [finite-...
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0answers
26 views

Stuck on combinatorial proof that $\binom{w}{p} \binom{p}{m} = \binom{w}{m} \binom{w-m}{p-m}$ [duplicate]

I'm in an introductory discrete mathematics course and i'm having a lot of trouble with this proof. I'm not sure if i'm heading in the right direction or not but I used the fact that $\binom{n}{k} = \...
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4answers
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Proving $\sum_\limits{n=1}^{\infty}\frac{1}{(n+z)^2}$ converges uniformly

Using the Weierstrass test show that the series $\sum_\limits{n=1}^{\infty}\frac{1}{(n+z)^2}$ converge uniformly on $E=Re(z)\geqslant 1$. This solution was given to me but I am not understanding ...
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Explaining the Result of an Equation

Let $Y$ be a closed subset of a Hilbert space $H$. Let $x \in H, z \in Y$. This equation comes from a proof in Lax' functional analysis book, and the other parts of the proof are not relevant: $$ 2tRe(...
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3answers
53 views

Please help me understand the following notation

Can someone kindly tell me the meaning of the following notation: A book defined the following matrix $(a_{ij})_{3\times 3}$ : $a_{ij}=\begin{cases} d_{ij}& i\neq j\\d_{ii}+\sum_{j=1}^3 d_{ij}&...
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1answer
25 views

Non-intuitive inverse function steps

In my precalculus book (open source Stitz and Zeager) we are trying to find the inverse function for $x^2$ - 2x + 4. (page 390). The book has steps in here that just don't make sense to me. Here ...
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2answers
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If $\limsup \frac{a_{n+1}}{a_n}<1$ does $\sum a_n$ converge even if $\lim\frac{a_{n+1}}{a_n}$ does not exist?

I was wondering why the Ratio test has the $\lim$ sign and the root test the $\limsup$ sign. Quotient test: $\lim|\frac{a_{n+1}}{a_n}|<1\Rightarrow \sum a_n$ converges. Ratio test: $\limsup \sqrt[...
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1answer
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How is this optimal value derived?

$\min (c^Tx) \text{ subject to } a^Tx \le b$ if $c = \lambda a$ for some $\lambda \lt 0$. Then the optimal value is $c^Tab = \lambda b$ Can someone explain how the optimal value is derived ...
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1answer
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Additive Function is Continuous

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that for any $x,y\in\mathbb{R}$ $f(x+y)=f(x)+f(y)$. Prove/Disprove $f$ must be continuous. Proof We have $\lim_{x\rightarrow a}f(x)=f(a)$ ...
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1answer
41 views

Doubt regarding Compactness argument used in Proof

I was reading Brian C Hall Lie algebra In that Book I come across following theorem. Notation Author mentioned that by standard argument of compact we can extract sequnce such that $A_{t_{k-1}^{-...
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0answers
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Point-set topological proof of Brouwer's fixed point theorem

I have tried to understand the point-set topological proof of Brouwer's fixed point theorem presented in Cou11. But I couldn't clarify some parts. Here are the theorem and its proof. Theorem: There ...
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0answers
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Proving that occupation measure of Brownian Motion is absolutely continuous almost surely

I am reading the section on occupation measures from Morters and Peres. I need some help with the following. $\{B(t):t\geq0\}$ denotes the standard Brownian Motion on the probability space $(\Omega,\...
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1answer
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Doubt in understading Proof of Matrix lie group and Lie algebra locally homemorphic

I was reading Brian C Hall Lie Group book In that I encountered following proof . I understand Whole proof. But have one doubt Why Auther take Orthogonal complement into consideration As I think ...
2
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2answers
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Confusion about a proof about subgroups of dihedral groups

This article shows that every subgroup of $D_n = \langle r, s \rangle$ is cyclic or dihedral. Theorem 3.1. Every subgroup of $D_n = \langle r, s \rangle$ is cyclic or dihedral. A complete listing ...
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Group proof/disproof v2 [duplicate]

How do I prove or disprove that, for a group $G$, if $a,b\in{}G$ and $aa_0=b$, then $a_0\in{}G$? Edit: I sillily said $b\in{}G$ instead of $a_0\in{}G$ originally. This is the reposted version written ...
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2answers
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Group proof/disproof [on hold]

How do I prove or disprove that, for a group $G$, if $a,b\in{}G$ and $aa_0=b$, then $a_0\in{}G$? Edit: I sillily said $b\in{}G$ instead of $a_0\in{}G$ originally.
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1answer
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Why must all co-efficents $c_j$ be positive in the integration of step functions?

I am working through a proof of the following theorem: 'If $f$ and $g$ are step functions having $f(x) \geq g(x)$ for all real values $x$, then $\int f \geq \int g$. So far I understand and thus ...
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0answers
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For gaussian process $X_t $ the best prediction is linear

I want to understand the proof for this Theorem: For gaussian process $X(t), t\in I $ the best prediction is linear. Proof: We only show the theorem for $ J \subset I $ with $\vert J \vert < \...
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0answers
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Cyclotomic polynomial identity proof with primes.

If p is prime, show that $\Phi_{p}(x^{p^{k-1}})=\Phi_{p^{k}}(x)$. Here is my attempt: $x^{p^{k}}-1=\prod_{d|p^{k}}\Phi_{d}(x)=\prod_{i=1}^{k}\Phi_{p^{i}}(x)=\Phi_{p^{k}}(x)(\Phi_{p}\Phi_{p^{2}}\...
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1answer
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Theorems & Proof Corrections [discrete mathematics]

So I've recently started a new chapter in my discrete mathematics course on proof methods. I have come across this problem in my textbook but I'm having a very hard time understanding where to start ...
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1answer
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Proof that $\sum\limits_{i=1}^\infty \frac{i}{(i+1)!} = 1$ [duplicate]

I came across this result randomly and am quite sure it's right. Is there any way to prove it rigorously? The numerator always seems to be one less than the denominator. Thanks!
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1answer
37 views

Cartier divisor example in Harthsorne

This question is about example 6.11.4 in Chapter II of Hartshorne. The example is about computing the Cartier divisor class group of the cuspidal cubic curve $y^2z = x^3$ in $\mathbb{P}_{k}^{2}$. He ...
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1answer
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Proof explanation of $P(p^e) \cong \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}/(p^{e-1})\mathbb{Z}$.

The following is from Classical Theory of Algebraic Numbers by Paulo Ribenboim : $P(p^e)$ is the set of all nonzero residue classes a modulo m, where gcd(a, m) = 1. My question underlined and in ...
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2answers
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Why does $a^{m_1}=a^{m_2}$ imply $a^{m_1-m_2}=e$?

I was reading this answer. I understand almost all of it. However, there is still one thing that continues to puzzle me. How should I prove for sure that, in this example, if $m_1\neq m_2$ and $a^{...
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2answers
29 views

Rational Equations: What are Excluded Values?

I have an example rational equation in a textbook which I was able to solve: $$\dfrac{3}{x-6} = \dfrac{5}{x}$$ Least common denominator is : $x(x-6)$ so: $$\frac{x(x-6)3}{x-6} = \dfrac{x(x-6)5}{...
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A proof of open mapping theorem?

I understand the proof before the circled lines, but why do we want the roots of the equation $f(z)=a$ to be simple for $a\not = w_0$? Why do we know we can choose $\epsilon$ so that $f'(z)$ does not ...
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1answer
31 views

Composite Moduli

The following passage can be found in Burton's Elementary Number Theory: "Assume that $a$ is odd. Because the square of any odd integer is congruent to 1 modulo 8, we see that for the congruence $x^2 ...
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Maths and Computing - prove that max length for base64 string is 12 [on hold]

Lets say I have a byte array, of 8-elements. //syntax c#: byte[] byteArray = new byte[] { 1, 2, 3, 4, 5, 6, 7, 8 }; Since a byte MAX is 255 in decimal equivalent, I can initialise my array to its ...
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1answer
29 views

Understanding limit representation of the exponential function

This is probably an easy question but I'm having trouble proving the following, I am lacking some mathematical knowledge. We know that... . Based on this we can say... . I don't want to just ...
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2answers
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Explaining the Proof of Schwarz Inequality for Scalar Product in a Vector Space

Let $\langle x,y\rangle$ be the scalar product of $x$ and $y$ in a linear space $X$ over either $\mathbb{R}$ or $\mathbb{C}$. This scalar product satisfies the three properties: Bilinerity/...
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3answers
96 views

Proving the limit $\lim\limits_{x\to \infty }\left(\sqrt{x}\left(\sqrt[x]{x}-1\right)\right)=0$

The solution were so messy that there was no way that I could came up with it on my own. Although the binomial Expansion seems reasonable the rest seems so forced because the solution required an ...
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0answers
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Prove that every simple connected graph with $2k$ edges can be partitioned in paths with 2 edges

I need to prove the theorem on title by induction in $k$ with $k>1$. To add I must show if the theorem holds if the simple graph is not connected. To show what happens when it is not connected I ...
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1answer
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A somewhat “familiar argument” about Artin's lemma

S. C. Newman proves in his book "A Classical introduction to Galois Theory", Chapter 9, Page 154, a lemma by Artin : I don't understand this "familiar argument" providing that $f\in K^H[x]$. I guess ...
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Can someone explain to me the jump between steps in the bottom two lines of this proof (not yet totally complete)?

Suppose we have a region $G$ that is bounded by the straight lines $x=a$, $x=b$, $y=c$ and by an arc $y = f(x)$ (which lies above $y=c$) where $a \leq x \leq b$. If $f$, $P(x,y)$ and $Q(x,y)$ are all ...
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3answers
55 views

If two sets are equivalent is it ok to write they are equal

I am trying to understand the proof of the following theorem: In the very first line of the proof, it is written that "It suffices to assume that $A_i=\mathbb{N}$" But I do not understand why....
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1answer
38 views

Recurrence of Brownian Motion

I am reading a proof of recurrence of Brownian Motion from the book of Morters and Peres. I have a question about a particular step in the proof of neighborhood recurrence for Brownian Motion in ...
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0answers
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Proof for upper bound of function using an approximation through Squeeze Theorem.

I am trying to turn $n_0log_2(n_0)$ into an approximation of $nlog_2(n)$ using the following statement: $$n \geq n_0+1 $$ For a bit of context, $n$ is the number of nodes on a minimum spanning tree [...
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Suppose that f : [a, b] → R is continuous and that f([a, b]) ⊂ [a, b]. Prove that there exists a point c ∈ [a, b] satisfying f(c) = c. [duplicate]

Suppose that $f : [a, b] \to \mathbb{R}$ is continuous and that $f([a, b]) \subset [a, b]$. Prove that there exists a point $c \in [a, b]$ satisfying $f(c) = c$. (If either $f(a) = a$ or $f(b) = b$ ...
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1answer
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Question on how to prove that a set has one-to-one correspondence with the set of positive integers

I am having a hard time understanding how to prove a question such as the following Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably ...
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1answer
40 views

Proof explanation of baby Rudin theorem 5.12

I'm having trouble understanding the proof for the intermediate value theorem for derivatives. Here is the thereom from Rudin: Suppose $f$ is a real differentiable function on $[a,b]$ and suppose $...
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1answer
53 views

What is meant by “an easy induction” in the solution to this problem?

I'm given the following problem (Putnam 1984/A2): Evaluate $$\sum_{n=1}^{\infty}\frac{6^n}{(3^{n+1}-2^{n+1})(3^n-2^n)}$$ The solution I've been able to find suggests that performing an 'easy ...
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2answers
68 views

$\{x\in A: x\not\in x\}$ is not a member of $A$

For any set $A$, the set $r(A):=\{x\in A: x\not\in x\}$ is not a member of $A$. It follows that the collection of all sets in not a set. The proof goes like this. By the separation axiom, $r(A)$ is a ...
0
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1answer
39 views

Universal set proof [discrete mathematics]

I've come across challenge proof question in my discrete mathematics textbook that I'm trying to solve for practice but unfortunately it does not have a solution. Any help with a reasonable ...