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.
12,450 questions
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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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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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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). ...
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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 $F$ ...
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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 $x^2$ ...
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Why is $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$ and not $\dfrac{\text{adjacent}}{\text{hypotenuse}}$? [closed]
Question:
I'm trying to understand why the sine ratio in a right triangle is defined as
$$
\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}},
$$
while the cosine ratio is
$$
\cos\theta = \frac{\...
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A proof of Lindenstrauss
I am trying to understand an argument by Lindestrauss in his paper LIPSCHITZ IMAGE OF A MEASURE-NULL SET
CAN HAVE A NULL COMPLEMENT* and there is one passage I do not understand. I believe that he ...
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Inductive proof of necessary and sufficient conditions for diagonalisability
I am very confused by the following highlighted lines in a proof of necessary and sufficient conditions for diagonalisability in Linear Algebra Done Right (4th ed.), Axler S. (2024).
Questions.
How ...
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Technical step in the proof of Linnik's theorem in Iwaniec-Kowalski (18.82)
Going through the proof of Linnik's theorem in Iwaniec and Kowalski's Analytic number theory, I came across an affirmation I don't really understand. On Page 440, starting from the explicit formula ...
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On finding the values of $ \theta$ for which $ \cos \theta \in \mathbb{Q} $
British Mathematical Olympiad Round 1 2008-09
Determine the sequences $a_0,a_1,a_2,\dots$ which satisfy all of the following conditions:
(a) $a_{n+1}=2a_n^2-1$ for every integer $n\ge 0$,
(b) $a_0$ ...
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How to combine the difference of two integrals with different upper limits?
The following link describes the Integral Comparison Test and its proof, along with this diagram which I understand.
This next theorem (and its proof) is what I am trying to understand:
There is ...
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Reflection of a Curve about a Line
so I was going through the following proof on how do we derive the new coorinates of the point on the reflected curve about a Line, and I couldn't understand one thing,How can we write $ x_f = x_0 + ...
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Understanding the solution of the transport equation
I would like to know how the following transport equation is solved:
$$ u_t+b \cdot \nabla_x u =f \text{ on } \mathbb{R}^n \times (0, \infty) \\
u=g \text{ on } \mathbb{R}^n \times {0},$$
where $b \in ...
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Understanding the proof of Dirichlet's test for convergence - part 2.
In the previous part to this question
I have the following proof of Dirichlet's convergence test:
Theorem (Dirichlet’s Test).
If the sequence $\{a_n\}$ is monotonically decreasing with $a_n \to 0$ as ...
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Classical comparison theorems for geodesics starting from a submanifold whose sectional curvature is bounded from below
Let $M$ be a 3-dimensional Cartan-Hadamard manifold (complete, simply connected, with nonpositive sectional curvature).
Let $S \subset M$ be a $C^{1,1}$ surface enclosing a domain $E$, and let $D_0$ ...
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Need help understanding the proof of the 'ratio test for sequences' (not series)
I think the parts boxed in red for the following 'ratio test for sequences' are incorrect:
Theorem:
Let $x_{n}$ be a sequence of positive numbers and let
$$L = \lim_{n \to \infty} \frac{x_{n+1}}{x_{n}}...
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Proof of deduction with only atomic consequences of $\bot_C$
I am reading Dag Prawitz's "Natural Deduction: A Proof-Theoreritcal Study" and am confused by the proof of Theorem 1, Chapter 3 that:
If $\Gamma \vdash_{\mathsf{C}'} A$, then there is a ...
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Switching integral and derivative: a specific case
Let $X=(X_1, \ldots, X_N)$ and $Y = (Y_1, \ldots, Y_N)$ be independent zero-mean Gaussian random vector. Let $Z(t) = \sqrt{1-t} X+\sqrt{t} Y$ for $t\in [0,1]$. Let $F \colon \mathbb{R}^N \to \mathbb{R}...
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What happened in the solution after writing the Mayer-Vietoris sequence?
I have been reading the first solution for the following question here:
How to use Mayer-Vietoris to show $\chi(X)=2\chi(M)-\chi(\partial M)$ where $X$ is the double of $M$?
Here is the first solution ...
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"Without loss of generality" in reference to division of inheritance
I am in the process of solving a warm-up problem (not graded) for a course I'm hoping to self-study this semester. I believe I have a solution sketch, and several older posts like here and here were ...
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Clarification on a passage of Theorem 3.3 of Milnor's Characteristic Classes
Let $\xi \subset \eta$ be a subbunlde of a vector bundle $\xi$ and suppose that $\eta$ is equipped with a metric. We define $\xi^\perp$ to be the orthogonal complement of $\xi$ in $\eta$.
So have a ...
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Explanation of a lemma on Hirsch’s Differential Topology.
On page 126 of Hirsch's Differential Topology book, he tries to prove the following lemma:
Lemma 1.7. Let $M$ and $N$ be $(n + 1)$-dimensional and $n$-dimensional, respectively, oriented manifolds ...
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Flawed "uniqueness" part of Picard-Lindelöf proof?
Edit: I think Dermot Craddock has provided a correct answer in the comments, but meanwhile I have found there is another (similar) answer here. I'm still a little confused and hope that others comment ...
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How is the strong Markov property applied here? (From chapter 1, §12, 'Probability-1' by Shiryaev.)
$\textbf{Proposition:}$ Let $\xi=(\xi_0,\ldots,\xi_n)$ be a homogeneous Markov chain with transition matrix $\|p_{ij}\|$, and let $$f^{(k)}_{ii}=P\{\xi_k = i,\xi_l \neq i, 1 \leq l \leq k-1 \vert \...
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Clarification about a particular step in a proof of the Kolmogorov-Arnold representation theorem
I am reading this proof of the Kolmogorov-Arnold representation theorem, first this sentences:
By plotting out the entire grid system, one can see that every point in $[0,1]^2$ is contained in $3$ to ...
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