Questions tagged [proof-explanation]
For posts seeking explanation or clarification of a specific step in a proof. "Please explain this proof" is off topic (too broad, missing context). Instead, the question must identify precisely which step in the proof requires explanation, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplicate questions.
12,456 questions
-2
votes
1
answer
37
views
Need help understanding a particular step in the proof of Cauchy's root test.
I am questioning a particular step of the solution presented to the following question:
Cauchy’s root test for convergence states the following: Given a series $\sum_{k=1}^\infty a_k$, define
$$\rho=\...
- 429
1
vote
0
answers
48
views
Any easier ways to prove an explicit form for a generated $\sigma$-algebra besides transfinite induction?
This question is a follow-up to an answer to a previous question, and motivated by my laziness in not wanting to learn about transfinite induction or how to write proofs using transfinite induction ...
- 1,716
3
votes
1
answer
181
views
I.N.Herstein "Topics in algebra" Sec. 1.1 The Set theory
I was recently reading I.N.Herstein's "Topics in algebra" and stumbled across interesting proposition and it's proof:
For any three sets, $A, B, C$ we have:
$$A \cap (B \cup C) = (A \cap B) ...
- 43
0
votes
1
answer
43
views
Understanding why the existence of an isomorphy of a finitely presented group to another one implies the relators is a finite set
I am currently taking a course on Free groups and we have the following proposition from B. Neumann, 1937:
if $ G = ⟨x_1,...,x_n|r_1,...,r_m⟩ = ⟨y_1,...,y_k|S⟩$, then there exists a finite subset $S_0=...
- 55
1
vote
0
answers
63
views
About Riesz representation theorem for $L^p$ spaces
First we did some reduction to only consider positive forms on $L^p(\Omega)$ with $\Omega$ a set of finite measure.
In the proof that we have been presented in class for this theorem, when we consider ...
- 11
3
votes
1
answer
85
views
+50
Is a collinearity step missing in this Miquel point proof?
Problem:
Solution:
Question: The problem and solution are taken from the book A beautiful journey through olympiad geometry. The problem is from the chpater $19$, complete quadrilateral. In the ...
- 127
1
vote
0
answers
58
views
Proof of $σ$-Additivity of Lebesgue Measure on Half-Open Rectangles.
I'm studying the proof that the volume function on half-open rectangles in $\mathbb{R}^n$ is a premeasure, specifically the inductive step for $\sigma$-additivity. The proof uses induction on the ...
- 53
3
votes
1
answer
73
views
Clarification on a step in the proof of the Kolmogorov extension theorem
This post is related to: Stuck in Tao's proof for Kolmogorov extension theorem
(Kolmogorov extension theorem) Let ${((X_\alpha,{\mathcal B}_\alpha),{\mathcal F}_\alpha)_{\alpha \in A}}$ be a family of ...
- 1,869
1
vote
1
answer
253
views
Compatibility of $\overline{d}$ with the completion topology $\overline{G}$
I am trying to understand a detail in the proof of Theorem 2.1.3 in Gao's Invariant Descriptive Set Theory.
The context is as follows:
$G$ is a topological group equipped with a left-invariant metric $...
- 781
2
votes
1
answer
88
views
Theorem 5, Chapter 7 in Chung (leading up to De Moivre-Laplace theorem)
I'm studying the below theorem from Elementary Probability Theory With Stochastic Processes by Chung. The proof is also shown in this answer. Cf. also a proof on Wikipedia. The proof in the book is ...
- 1,618
1
vote
0
answers
79
views
Unclear step in van Maaren's theorem proof by Schechter
I am trying to disentangle the proof of Brouwer's fixed point theorem via van Maaren's geometry-free Sperner lemma in Eric Schechter's Handbook of Analysis and its Foundations (sections 3.28-3.37). ...
- 61
0
votes
0
answers
50
views
Understanding the application of FTC in the proof that $\dim \left( T_{p}M \right) = n$
Consider the following proof I am trying to break down
There are two things I don't understand in the proof.
I do not understand the way the author uses the Fundamental Theorem of Calculus. I know ...
- 205
-2
votes
1
answer
142
views
What is Gödel's argument for why his proof for a single system applies to all systems
I'm having trouble understanding how Gödel extrapolates from a consistent formal system to any formal system.
For reference, his First Incompleteness Theorem states:
Any consistent formal system $F$ ...
- 97
2
votes
6
answers
1k
views
Example that the union of an infinite number of closed sets is not necessarily closed.
Example. Let $I_n = [1/n, 1]$, which is clearly closed, and consider
$$S=\bigcup_{n=2}^{\infty}I_n=[1/2,1]\cup[1/3,1]\cup[1/4,1]\cdots\tag{1}$$
This is the set
$$S=\bigg\{x\bigg\lvert x\in \mathbb{R},\...
- 581
1
vote
2
answers
136
views
Proof writing standard: English vs Symbols. What's better? [closed]
I’m new to proof writing. For a general proof, I’ve come across books writing proofs by use of formal grammar and math. Take this common textbook example,
(1) Proposition: If $x$ is even, then $x^2$ ...
- 61