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.
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Robinson's word vs Rotman's word: can the second be derived from the first?
In "A Course in the Theory of Groups" written by Derek J.S. Robinson it is done as follows.
Let be $X$ a set. Choose a set disjoint from $X$ with the same cardinality (see here for details):...
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Why is $\pi_{0} \left(G \right)$ a discrete and countable Lie group
At the moment I am taking a first course about Lie groups, and am using these notes. At the bottom of page $20$ the author introduces the notation $gG^{\circ}$ for the connected component of $g\in G$ ...
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$(X, \Sigma, \mu)$ finite measure space and $(f_n)_{n\in\mathbb{N}}$ is sequence in $L^p(X)$. If $f_n \to f$ uniformly on $X$, then $f \in L^p(X)$.
Help in understanding a proof written by a teacher on the following theorem.
Let $(X, \Sigma, \mu)$ be a finite measure space and let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions in $L^p(X)$.
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Step in Brezis' proof of chain rule for Sobolev spaces
I'm going through the proof of Corollary 8.11 in Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations which states:
Let $G \in C^1(\mathbb{R})$ be such that $G(0) = 0$, and ...
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Help understanding this Carter-Wegman proof on the lower bound of collisions in a hash family
In this paper: https://www.sciencedirect.com/science/article/pii/0022000079900448
Proposition 1 explains how the probability of a collection of hash functions $H$ with each $f\in H$ mapping $A$ to $B$ ...
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Case where infeasibility comes from $Ax=b$ unmet in standard polyhedron not handled in this proof?
In linear optimization, a solution that is not feasible can be so because it violates an equality constraint, isn't it? Be it an equality constraint from a non-standard polyhedron or an equality ...
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Understanding 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
$$\rho=\...
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Any easier ways to prove an explicit form for a generated $\sigma$-algebra besides transfinite induction?
This question is a follow-up to an answer to a previous question, and motivated by my laziness in not wanting to learn about transfinite induction or how to write proofs using transfinite induction ...
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I.N.Herstein "Topics in algebra" Sec. 1.1 The Set theory
I was recently reading I.N.Herstein's "Topics in algebra" and stumbled across interesting proposition and it's proof:
For any three sets, $A, B, C$ we have:
$$A \cap (B \cup C) = (A \cap B) ...
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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=...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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