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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15 views

Show that $m^{*}([a, b]\backslash G)=b − a- m^{*}{(G)}$

In the proof of Theorem 1.23 in Real Analysis by Bruckner it is used the following without a proof : Let $G$ be an open subset of an interval $[a, b]$ and write $K =[a, b]\backslash G$. Then $m^{*}{(...
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Understanding the proof of the condition number of a matrix.

I am trying to understand how the condition number of a matrix is deduced from a linear system of equation. I have a system of equations $A \vec{x} = \vec{b}$ to which we introduce certain ...
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1answer
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How do I prove that the cross product follows the right hand rule for vectors in the xz plane?

Thanks to this wonderful article I have been able to prove that the cross product follows the right hand rule when the cross product has a non-zero z component. In this case we show that the z ...
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The logistics of a V to W linear transformation [closed]

If T is a linear transformation from V to W, and w is not in Range(T), does that 2w is also not im Range(T)
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How to show that semi-direct product is isomorphic to a direct product

On page 182 of Dummit and Foote Third Edition the following is stated during their classification of groups of order 30. I have worked through the entirety of the process, however I am confused at the ...
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4answers
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How to prove that $8|(4k^2 + 4k)$

$a|b$ iff there exists $c$ in the integers such that $b=ac$ but I don't know how to apply this definition to prove the above
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Problem with a derivation using the continuity equation

Consider the continuity equation for an electron gas: $$\tag{1} \nabla \cdot\left[n(\boldsymbol{r}, t) \frac{\partial}{\partial t} \tilde{\boldsymbol{r}}(\boldsymbol{r}, t)\right]=-\frac{\partial}{\...
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1answer
30 views

show that sequence of functions $f_n(x) = (nx^2+1)/(nx+1)$ converges uniformly on $[1,2]$

I have found out that this sequence of functions approaches to $1$ at $x=1$, approaches to $2$ at $x=2$, approaches to $x$ when $1<x<2$. Now to prove the sequence converges uniformly in $[1,2]$, ...
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2answers
36 views

Strong Induction proof : $x_k = 1/2(x_{k+1}+x_{k-1})-1$ holds for all integers $k∈Z_{≥0}$

I have just started to learn how to do strong induction. I am struggling to fully understand how to properly do it though. I am trying to work on this exercise where I must prove $x_k = 1/2(x_{k+1}+x_{...
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1answer
68 views

Using binomial theorem to express $ \frac{1}{\sqrt{5}} \ [((\sqrt{5}/2)+(1/2))^{10} - (-(\sqrt{5}/2)+(1/2))^{10}]$ as a single finite series

I am trying to write $$\frac1{\sqrt5}\left[(\frac{\sqrt5}2+\frac12)^{10} - (-\frac{\sqrt5}2+\frac12)^{10}\right] $$ as a single finite series of the form $\sum^{10}_{j=0}a_j$, where $a_j$ depends on $...
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Explanation of the proof of $\phi(n)\cdot \frac{n}{2}=\sum_{a\in R}a$

In my number theory textbook, there is a theorem about Euler's Phi Function it says: A Theorem: If $R$ is a reduced residue system modulo $n$: $R=\{a\in C \mid \gcd(a,n)=1\}$ Where $C$ is a complete ...
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1answer
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in barycentric coordinates why does $[PBC] = x[ABC]$?

From the Euclidean Geometry in Mathematical Olympiads written by Even Chan, there is a chapter about barycentric coordinates. It is said in the chapter that Barycentric coordinates are also sometimes ...
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1answer
135 views

What is the motivation behind the steps in this 'simple' proof that $\pi$ is irrational?

In $1947$, Ivan Niven published A Simple Proof that $\pi$ is irrational, which only requires knowledge of elementary calculus to understand. Since then, many variations of this proof have been ...
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Proving whether a function is injective…

I am confused on how to prove this function. It is $f(x)$ but within its set, it involves $y$. For example, $f(x)=\{y \in \mathbb R:....\}$ (where the .... is an equation) To prove its injectivity, ...
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32 views

How to prove that $(X^y - X) \bmod 3 = 0$ always when $y$ is odd and $y > 1$, and $X >1$? [duplicate]

My question is quite simple: how to prove that $(X^y - X) \bmod 3 = 0$ when $y$ is odd and $y > 1$, and $x $ is an integer greater than $1$? Examples: $(2^3-2)/3=2$ $(11^3-11)/3=440$ $(7^5-7)/3=...
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Determining continuity of a function

I'm supposed to determine continuity of a function at different points $f'(x)=(x-m)^{2k} (x-n)^{2n-1}$ where $m<n$ and $k$ is a positive integer The answer says $m$ is neither maxima nor ...
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1answer
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Understanding proof of $\operatorname{PSL}_2(\mathbb{F}_2)\cong S_3$ and $\operatorname{PSL}_2(\mathbb{F}_3)\cong A_4$

Given proof: Consider the action of $\operatorname{PGL}_2(\mathbb{F}_q)$ on the projective line $\mathbb{P}^1(\mathbb{F}_q)$. This action is faithful and 3-transitive, so $$ \phi:\operatorname{PGL}_2(\...
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1answer
28 views

Proof of $\ln(p_k(x))=\delta_k(x)$

In "An introduction to Statistical Learning in R" by James, Witten, Hastie, and Tibshirani, on page 139-140, in the section concerning Linear Discriminant Analysis for p=1, assuming $f_k(x)\...
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Metric space if and only if proof

Given that $1 \leq a \leq \infty$ and $1 \leq b \leq \infty$, consider the two metrics on $\mathbb{R}^n$: $d_a(x, y) = (\sum_{i = 1}^{n} |x_i - y_i|^a)^{\frac{1}{a}}$ and $d_b(x, y) = (\sum_{i = 1}^{n}...
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Show $\{p_n\} \to p \implies \{\operatorname{diam}E_N\} \to 0$

Here's a little discussion from my book that I rewrote so I can understand it better. Part 1 Part 2 Here below is the discussion linked above that I rewrote. Please see if it makes sense. Thanks. Let ...
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109 views

Questions about Rudin's proof of Lebesgue's Monotone Convergence Theorem

I just finished working through Papa Rudin's proof of Lebesgue's Monotone Convergence Theorem, and I have some questions: Why are each $E_n$ measurable? How is $(6)$ concluded, i.e. $\displaystyle \...
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1answer
18 views

Evaluating $\int_{E_{ij}} (s+t)\ d\mu$ in Proposition $1.25$ (Rudin's RCA)

I am unsure how $$\int_{E_{ij}} (s+t)\ d\mu$$ has been evaluated in Proposition 1.25, and I've attached a picture of the same for context: (Please scroll down to below the image to see my question and ...
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3answers
41 views

$\int_0^a \mid f(t) \mid dt \rightarrow 0 \Rightarrow f = 0 $?

I was reading a proof and I stucked at some point that I don't understand. Let's consider norm : $$||f||_1 = \int_0^1 |f(t)| dt$$ Now I will cite a part of the proof that I don't get I don't ...
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1answer
33 views

A function $f: \mathbb{R}^n \to \mathbb{R}^m$ is differentiable if and only if its components are differentiable

I have a function $f: \mathbb{R}^n \to \mathbb{R}^m$ and need to prove that it's differentiable if and only if its component functions are differentiable. The definition I'm working with is far ...
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1answer
21 views

$|\int f \ d\alpha| \leq \int |f| \ d\alpha$ (Rudin 6.13)

I'm reading Rudin's proof of theorem 6.13, $$\left|\int_a^b f\ d\alpha \right| \le \int_a^b |f| \ d\alpha$$ and realizing there's a step I can't seem to understand. We let $c=\pm 1$ such that $$\left|...
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63 views

Proof that [0,1] is compact

Is this a valid proof that $K=[0,1]$ is compact? Suppose we had a open cover $\{G_a\}$ of $K$ that did not give a finite subcover. We split $[0,1]$ into $[0,1/2]$ and $[1/2,1]$. At least one of these ...
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2answers
41 views

Question on proof of Abel's limit theorem

Let be $\sum\limits_{k=0}^{\infty}a_kx^k$ a power series with $a_k\in\mathbb{R}$ , radius of convergence $0<R<\infty$ and assume that $\sum\limits_{k=0}^{\infty}a_kR^k$ exists. Then, Abel's ...
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3answers
80 views

Example of presentation of a group.

I’m reading up on presentations of a group. In here, the example is the dihedral group $D_8$, with generators rotation $r$ of order $8$, and flip $f$ of order $2$. In the construction of the ...
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1answer
27 views

Confusion on the proof of Haim Brezis Proposition 1.9 (application of Hahn-Banach Geometric Form)

I am reading Haim Brezis's functional analysis, and I am confused by on of his proof. Let $(V, \|\ \cdot\ \|)$ be a Banach space on $\mathbb{C}$ and let $V^{*}$ be its dual, endowed with the dual norm ...
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95 views
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A statement in the book “Differential and Integral Calculus, 6th ed” (Love and Rainville)

In page 285 of "Differential and Integral Calculus, 6th ed." by Love and Rainville, the example used in 'Substitution suggested by the problem' about plane areas is this: Find the area of ...
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1answer
34 views

Non-zero coefficient in a transcendence proof

I am studying the proof of the simple version of Lindemann's Theorem. Theorem. If $\alpha$ is a non-zero algebraic number, then $e^\alpha$ is transcendental. Apologies for the long read, but to ...
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1answer
46 views

Unions of closed sets

Let $\bar{A}$ be the closure of $A$. Let $B_n = \bigcup_{i=1}^{n}A_i$ for positive integer $n$. I want to show that $\bar{B}_n = \bigcup_{i=1}^{n}\bar{A}_i$. I am thinking if $x$ is a limit point of $...
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1answer
34 views

Question about the proof of the fact that subsequential limits are closed

Theorem: For any sequence $\{p_n\}$ in a metric space $X$, the set $E$ of all subsequential limits of $\{p_n\}$ is closed relative to $X$. I am not sure I follow the(?) proof of this theorem in my ...
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0answers
41 views

Conjugacy classes of the Dihedral group $D_n$ - Proof [closed]

I'm trying to understand this proof of the number of conjugation classes of the dihedral group $D_n$ and I can follow it up the part where it states how the conjugate class of x looks like. But I don’...
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1answer
25 views

Prove that the functions $h(x)=|f(x)|$ and $k(x)=\max\{f(x),g(x)\}$ are both continuous. [duplicate]

Here is the full question: Let $X$ be a topological space and $f,g$ two continuous functions $f:X\to \Bbb R$. Prove that the functions $h(x)=|f(x)|$ and $k(x)=\max\{f(x),g(x)\}$ are both continuous. I ...
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1answer
29 views

Modular arithmetic $a=bq+r$ [closed]

How do I show that if $r$ is the remainder when $a$ is divided by $b$ (e.g. $a = bq + r$) then $a\equiv r (\mod b)$? I assume I'm supposed to use the division algorithm and/or the quotient remainder ...
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1answer
36 views

$2^{nd}$ countable implies separable

I was looking at this proof that $2^{nd}$ countable implies separable. Could someone clarify if my understanding is correct? Suppose $X$ is $2^{nd}$ countable having countable basis $\beta=\{B_i|i \in ...
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1answer
57 views

Proof explanation for how this criteria shows exactness

This is about a proof from Homological Algebra on a Complete Intersection... by Eisenbud. Here, $A$ is a Noetherian commutative ring. To show exactness at $\bar F$, we must show that $\operatorname{Im}...
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Good filtrations on $A_n(K)$ modules

We are reading J. E. Björk's book: Rings of Differential Operators and we don't understand one step at Lemma 3.4: Let $\Gamma$ and $\Omega$ be two filtrations on the left $A_n(K)$-module $M$ and ...
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2answers
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Proof of Lemma 6.3 in Carothers' Real Analysis

Lemma 6.3 (Chapter 6 - Connectedness): Let $E$ be a subset of a metric space $(M,d)$. If $U$ and $V$ are disjoint, open sets in $E$, then there are disjoint open sets $A$ and $B$ in $M$ such that $U= ...
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1answer
29 views

Proof Explanation - $\Bbb R$ is connected, i.e. has no non-trivial clopen subsets

I know that $\Bbb R$ is connected iff it has no non-trivial clopen subsets. To show that $\Bbb R$ has no non-trivial clopen subsets, I came across the following solution which I need help in ...
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5 views

Does $n(u, v) · (w + αu + βv) = n(u, v) · w$ imply that the angle between the vectors is the same?

I'm trying to work my way through Gao's "A simple proof of the right hand rule". In this proof we have vectors $u$ and $v$ which form a plane, and $n(u,v)$ which is perpendicular to both ...
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56 views

Prove Prop 4.11

4.11 Proposition. Let $X$ be a set and let $\mathcal{S}$ be any collection of subsets of $X$ such that $X=\bigcup_{V\in\mathcal{S}}V$. Let $\mathcal{T}$ denote the collection of all subsets of $X$ ...
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2answers
74 views

Why is $0^0$ undefined and how would we graph this?

So I saw exponents like $3^0$ and $4^0$, etc which are all equal to $1$. And then soon I see that $0^0$ is not defined. I checked the graph of $x^0$ Then I decided to make some observations. We go ...
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24 views

Convex sets and empty interior

While self studying convex analysis i got a little stuck regarding one of the proofs given by the authors regarding the Corollary. Theorem Let $S$ be a convex set in $R^{n}$ with a nonempty interior. ...
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0answers
31 views

How to prove that all integers with more than 1 digits, minus the sum of their digits result in $\mod 3=0$? [duplicate]

I have ran a script on the first $100000$ natural numbers and noticed that for any Integer (with more than 1 digit) minus the sum of its digits will result in $\mod 3 = 0$ ($w$ Starting Integer) → ($x$...
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0answers
11 views

Relationship between matrix sum, ranks, and singular values in Eckart-Young-Mirsky proof

I've read the proof of the Eckart-Young-Mirsky theorem on WikiPedia (for Frobenius norm), and I understand all of it except this one step: $\sigma_1(A - A'_{i - 1} - A''_{j - 1}) \geq \sigma_1(A - A_{...
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1answer
23 views

Golden Mean - Prove, in general, that For all $\in N | τ^{n+1} = τ^n + τ^{n-1} $

Prove in general that For all $\in N | τ^{n+1} = τ^n + τ^{n-1} $ I've been trying to work on this problem over the past few days and I seem to be missing something. I know that $ τ = (1 + √5)/2$ I ...
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4answers
79 views

How to prove that $\neg (p \land \neg q) \lor (\neg p \land q) \equiv \neg p \lor q$?

This is what i got so far: (~pvq) v (q∧~p) ≡ I like using association property, but it's not possible in this case, because of (q∧~p), if I use Morgan's law on this part, this is what will happen (~...
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1answer
41 views

Show the intersection of 2 subspace topologies is a subspace

Background The following problem is out of my head. You can refer to Topology Without Tears by Sidney A. Morris for definitions: Theorem Let (X $\tau$) Y,Z $\subseteq X $ be a subspace of X with ...

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