# 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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### Doubt in a proposition from Fulton's Algebraic Curves, section 5.5 (some criteria for Noether's condition)

I was reading section on Max Noether’s Fundamental Theorem in Fulton's Algebraic Curves and came across the following proposition which gives some sufficient criteria for Noether's condition to hold (...
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### Questions on a proof on $p$-constrained groups

Theorem: Let $G$ be a group and $p \in \pi(G)$. Furthermore, suppose that $$\label{eq_p-constrained} C_{G/O_{p'}(G)}(O_p(G/{O_{p'}(G)})) \leq O_p(G/{O_{p'}(G)}).$$ If $P$ ...
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### Trouble understanding the proof of the theorem: If $f\in \mathfrak{a}$, then $m_i\in \mathfrak{a}$ for each $i$,

The following are from Froberg's Introduction to Grobner bases, Malik anad Sen's Fundamentals of Abstract Algebra Background Lemma: Let $I$ be a momomial ideal and $f\in K[x_1,\ldots,x_{n-1},x_n]$. ...
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### Proving $\lim_{y\to 0} y^2 \ln|yx^2|=0$ using sequences

I have the function $$f(x, y) = y^2 \ln|yx^2|$$ and I want to prove that $f(x, 0)$ goes to zero but using sequences. SO I thought about this: I choose $b_n = \frac{1}{n}$, and in general any $b_n$ ...
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### Explanation of an inequality in proof of nested intervals property

I am having trouble understand part of a proof of Nested intervals property: Let $F$ be an ordered field with the monotone sequences property. Let $I_1\supseteq I_2 \supseteq \cdots$ be closed bounded ...
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### $C[a,b]$ is dense in $L^p([a,b])$ from the fact that $C_c(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$

We know that the following theorem holds. Theorem. The space $C_c(\mathbb{R}^n)$ of the continuous functions with compact support is dense in $L^p(\mathbb{R}^n)$ for $p\in [1,\infty).$ I don't ...
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### Question to a solution to an invariance problem from Arthur Engel's book

I have a question with the answer Engel provides to the following problem: Three integers $a, b, c$ are written on a blackboard. Then one of the integers is erased and replaced by the sum of the other ...
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### Defining the completion of a group can be done only using Cauchy sequences

Let $G$ be a group, in Atiyah & MacDonald's Commutative Algebra it says that Assume for simplicity that $0\in G$ has a countable fundamental system of neighborhoods. The completion $\hat G$ of $G$...
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### Proof of Thompsons $A \times B$-lemma

(Auxiliary lemma) Let $G$ be a $\pi$-group and $a$ a $\pi'$-element acting on $G$. If $X$ is a subnormal subgroup of $G$ with $[a, X] = 1 = [a, C_G(X)]$, then $[a, G] = 1$. Hey guys, I am having a ...
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### A question in one step of proving Van Der Corput's Difference Theorem

In proving Van Der Corput's Difference Theorem, there's a lemma: Suppose $\left\{ z_{n} \right\}$ is a bounded complex-valued sequence. Then if for all $d \in \mathbb{N}$ , we have \begin{align*} \lim\...
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### Theorem 3.6.4 (Separable space) Kreyszig.

Kreyszig , in "Introductory Functional Analysis with Applications" , has this Theorem 3.6.4 (with Proof) concerning Separable spaces. Theorem. Let $H$ be a Hilbert space. If $H$ contains an ...
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### Proposition 8 Corollary 1, Section 5.7 of Hungerford’s Algebra

Corollary 1.9. Let $E$ and $F$ each be extension fields of $K$ and let $u\in E$ and $v\in F$ be algebraic over $K$. Then $u$ and $v$ are roots of the same irreducible polynomial $f \in K[x]$ if and ...
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### Projection into a Hilbert space with respect to an orthonormal sequence.

Let $H$ be an Hilbert space and let $(e_k)_{k\in\mathbb{N}}$ be an orthonormal sequence in $H$. We define $$V:=\overline{\text{span}(e_k)}$$ I must prove that for all $x\in H$ the projection onto $V$ ...
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Example. Let $F=\{E_1,E_2,\ldots,E_s\}$ be a family of subsets with $r$ elements of some set $X$. Show that if the intersection of any $r+1$ (not necessarily distinct) sets in $F$ is nonempty, then ...
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### Understanding the proof of Perron–Frobenius theorem

I reviewed the proof for Perron–Frobenius' theorem as stated in this article. In the proof, they define Q to be a positive orthant ( $Q:= \{ x \in \mathbb{R}^n: x \geq 0, x \neq 0 \}$) For an ...
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### Centered Subgaussian Variables have better Properties

I am trying to understand the following proof: Main Confusion: In particular, I am having a very hard time understanding the chain of inequalities in the proof for (3)': I think the first equality is ...
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### Understanding the proof for Properties of Subgaussian Variables

Here are the definitions, statements and the proof that I am stuck on: I am stuck on the last part of the proof where the author claims that setting $C = e$ automatically guaranties that (1) holds ...
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Find the volume of the solid inside the cylinder $x^2+y^2-2ay = 0$ and between the plane $z = 0$ and the cone $x^2+y^2 = z^2$. I tried solving this problem as follows: Equation of the cylinder $x^2+(y-... • 3,065 4 votes 2 answers 99 views ### Trying to prove monotone-sequences property ⇒ Archimedean property Monotone-sequences property ⇒ Archimedean property Today I just started learning this and having trouble understanding parts of the proof. Sorry if these are easy to some, I just couldn't find this ... • 95 6 votes 3 answers 186 views ### Prove that any sequence of five distinct integers must contain a 3-chain This task is from MIT OpenCourse Mathematics for CS 2010 course, problem set 2, exercise 1(d). I am aware that this question has already been asked several times previously on this platform. Yet, the ... • 86 0 votes 1 answer 44 views ### For a square matrix$A$, if$\dim(ker(A - 2I) = 3$, then its characteristic polynomial is of the form ($\lambda - 2)^3$* q ($\lambda$) For a square matrix$A$, if$\dim(ker(A - 2I) = 3$, then its characteristic polynomial is of the form ($\lambda - 2)^3$* q ($\lambda$) for some polynomial q. My approach: So we know that A has an ... • 157 3 votes 0 answers 37 views ### Norm surjective for unramifeid extension of local fields$L/K$I have a question about the argument used here in proof of Proposition 4.2.5.: For any finite unramified extension$L/K$of local fields, the map$\text{Norm}_{L/K}: O_L^* \to O_K^*$is surjective. ... • 7,499 0 votes 2 answers 42 views ### Axler Theorem 5.18:$\text{null}(T)$and$\text{range}(T)$are invariant under$T$. I am trying to understood Axler's proof of Theorem 5.18. It states that: if$T$is a linear operator from$V$to$V$and$p$is a polynomial with coefficients in the field$F$, then$\text{null} p(T)$... • 1,243 0 votes 1 answer 90 views ### Proposition 12, Section 5.6 of Hungerford’s Algebra Let$F$be a finite dimensional extension field of$K$and$N$a normal extension field of$K$containing$F$. The number of distinct$K$-monomorphisms$F\to N$is precisely$[F : K]_s$, the separable ... • 4,239 1 vote 0 answers 38 views ### Axler Theorem 5.17, part (b) I am trying to understand the proof of part (b) of Theorem 5.17 in Axler's Linear Algebra Done Right. He cites part (a) in his proof of (b), so I've written out the full theorem statement below.$\...
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I'm following a proof given by Spivak in his textbook "Calculus" and one of the steps seems slightly unjustified to me. Firstly, here's the theorem: If $f$ is a function defined on $(a,b)$ ...