Questions tagged [proof-explanation]
This tag is for readers who ask for explanation and clarification of some steps of a particular proof.
8,641
questions
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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^{*}{(...
0
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0answers
10 views
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 ...
0
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1answer
14 views
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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0answers
8 views
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)
0
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0answers
31 views
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
54 views
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
0
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0answers
24 views
Problem with a derivation using the continuity equation
Consider the continuity equation for an electron gas:
$$\tag{1}
\nabla \cdot\left[n(\boldsymbol{r}, t) \frac{\partial}{\partial t} \tilde{\boldsymbol{r}}(\boldsymbol{r}, t)\right]=-\frac{\partial}{\...
0
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1answer
30 views
show that sequence of functions $f_n(x) = (nx^2+1)/(nx+1)$ converges uniformly on $[1,2]$
I have found out that this sequence of functions approaches to $1$ at $x=1$, approaches to $2$ at $x=2$, approaches to $x$ when $1<x<2$.
Now to prove the sequence converges uniformly in $[1,2]$, ...
1
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2answers
36 views
Strong Induction proof : $x_k = 1/2(x_{k+1}+x_{k-1})-1$ holds for all integers $k∈Z_{≥0}$
I have just started to learn how to do strong induction. I am struggling to fully understand how to properly do it though. I am trying to work on this exercise where I must prove $x_k = 1/2(x_{k+1}+x_{...
1
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1answer
68 views
Using binomial theorem to express $ \frac{1}{\sqrt{5}} \ [((\sqrt{5}/2)+(1/2))^{10} - (-(\sqrt{5}/2)+(1/2))^{10}]$ as a single finite series
I am trying to write
$$\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 $...
0
votes
1answer
31 views
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 ...
1
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1answer
16 views
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 ...
4
votes
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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0answers
27 views
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, ...
0
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0answers
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=...
0
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0answers
22 views
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 ...
1
vote
1answer
24 views
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(\...
0
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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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0answers
38 views
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}...
0
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0answers
11 views
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 ...
2
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2answers
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 \...
0
votes
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 ...
0
votes
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 ...
0
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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 ...
0
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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|...
3
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0answers
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 ...
1
vote
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 ...
2
votes
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 ...
1
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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 ...
1
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0answers
95 views
+50
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 ...
2
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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 ...
1
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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 $...
2
votes
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 ...
1
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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’...
0
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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 ...
0
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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 ...
0
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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 ...
0
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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}...
1
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0answers
13 views
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 ...
4
votes
2answers
73 views
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= ...
1
vote
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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0answers
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 ...
0
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2answers
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 $X$ ...
3
votes
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 ...
-1
votes
0answers
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. ...
1
vote
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) → ($x$...
1
vote
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_{...
0
votes
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 ...
-2
votes
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
(~...
0
votes
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 ...
