Questions tagged [proof-explanation]

This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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Proof from Lang’s Basic Mathematics

Starting on page 11, I don’t understand the proof. If a, b are negative integers, then a + b is negative. Proof. We can write a = —n and b = —m, where m, n are positive. Therefore a + b = —n — m = —(n ...
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Meaning of symbol in probability theorem

What does the $\big|_{t=0}$ mean in this context?
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Spivak Calculus Ch. 11, Prob. 61a: $f$ diff in interval containing $a$, $f'$ discont. at $a$. Prove one-sided limits of $f'$ at $a$ cannot both exist.

The following is a problem from ch. 11 of Spivak's Calculus Suppose that $f$ is differentiable in some interval containing $a$, but that $f'$ is discontinuous at $a$. Prove the following: (a) The ...
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$f$ cont. at $a$, $f'$ exists in interval containing $a$ (except possibly at $a$), $l=\lim\limits_{x \to a^+} f'(x)$ exists. Does $f'(a)=l$?

This question regards the following theorem (as stated in Spivak's Calculus): Theorem 7: Suppose $f$ is continuous at $a$, $f'(x)$ exists for all $x$ in some interval containing $a$, except perhaps ...
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uniqueness of clutching decomposition Hatcher p.$47$ Lemma $208$

In the following accepted questions on mse (first and second) concerning K theory, in particular the uniqueness of the splitting given also in Hatcher p.$47$ Lemma $208$ is addressed by the same ...
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Mistake in proof of "double derivative test" in calculus textbook

I'm currently studying for a semester test in advanced calculus, and one of the topics covered is finding the local minima and maxima of a 3 dimensional surface. The first theorem that was proved was ...
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Prove that every cycle graph $C_n$ has $n$ edges

I need to prove this directly and by induction. I do not even know where to start. Question: A cycle graph $C_n$ is a connected graph with $n$ vertices, such that each vertex is adjacent to exactly ...
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Fredholm Index of Toeplitz operators with invertible and continuous symbol

I'm working through the following proof in C* algebras by Murphy, and I'm stuck on a step in the proof. For reference, $\epsilon_n = z^n : T \longrightarrow \mathbb{C}$, and $\Gamma = \text{span}(\...
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$x=Px+Qx$ unique decomposition in Hilbert space

Problem: Let $M$ be a closed subspace of Hilbert space $H$. Then every $x \in H$ has unique decomposition $x = Px + Qx$ where $Px \in M$, $Qx \in M^{\perp}$ . I don't understand: When proving that our ...
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Spivak's Calculus, Ch 11, problem 59: $f$ has property $(f')^2=f+\frac{1}{f^3}$, find formula for $f''$ in terms of $f$.

The following is a problem from chapter 11, "Significance of the Derivative", from Spivak's Calculus Redo problem 10-18(c) when $$(f')^2=f-\frac{1}{f^2}\tag{1}$$ Here is problem 10-18(c) ...
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Proof of Łoś-Tarski theorem: explanation of the obtained contradiction

I'm going through the following document about model theory: https://webspace.science.uu.nl/~ooste110/syllabi/modelthmoeder.pdf in which a proof of the Łoś-Tarski preservation theorem is given. I ...
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Spivak's Calculus, Ch. 11, "Significance of the Derivative", prob. 58: Prove $f'$ increasing then every tangent line intersects graph of $f$ only once

The following is a problem from ch. 10, "Significance of the Derivative", from Spivak's Calculus Prove that if $f'$ is increasing, then every tangent line of $f$ intersects the graph of $f$ ...
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Spivak's Calculus: Understanding proof of alternative form of l'Hôpital's Rule

The following problem is from chapter 11, "Significance of the Derivative", of Spivak's Calculus. There is another form of l'Hôpital's Rule which requires more than algebraic manipulations: ...
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An example showing that a quotient space of an hausdorff space is not hausdorff.

Let be $\mathscr P$ the partition of $[0,1]^2$ defined through the position $$ \mathscr P:=\big\{\{x\}\times[0,1]:x\in\Bbb Q\big\}\cup\big\{\{(x,y)\}:(x,y)\in \mathbb R \setminus \mathbb Q\times[0,1]\...
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Understanding a 'geometrical proof' of irrationality of √2

I had been having trouble understanding a proof of the irrational nature of √2. I found this proof in the first page of the foreward to 17 theorem provers of the world where a 'geometrical proof' (is ...
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Doubt from the proof that the sequence space $l^2$ is a Hilbert space from Kreszig book

This is the proof given in Kreszig book. 3.1-6 Hilbert sequence space $l^2$. The space $l^2$ (cf. 2.2-3) is a Hilbert space with inner product defined by $$ \langle x, y\rangle=\sum_{j=1}^\infty \...
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If $\tau ^{(n)}\downarrow \tau $ then $\{X_{\tau ^{(n)}}\}_{n\in \mathbb{N}}$ is uniformly integrable

Reading in the book of Bhattacharya and Waymire of probability theory I find the following assertion: Let $\{X_t\}_{t\in [0,T]}$ a right-continuous martingale adapted to a filtration $\{\mathcal F_t\}...
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In the proof of construction of canonical module

I'm reading David Eisenbud's Commutative Algebra, p.539, Theorem 21.15 : I'm trying to understand the underlined statement. Why is it true? My first attempt is, Question 1. Let $I:=\operatorname{...
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3 votes
2 answers
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How to rigorously fill in certain steps in the proof of L'Hôpital's Rule as it appears in Spivak's Calculus?

The following is L'Hôpital's Rule as it appears in Spivak's Calculus Suppose that $\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} g(x)=0$ and suppose also that $\lim\limits_{x \to a} \frac{f'(x)}...
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Derivation of an upper bound

This is question 24 of this collection of dice problems with solutions, I couldn't understand this $$\begin{align} \frac{1+5^n+\max\{1+6,1+C\log(n-1)\}\sum_{j=1}^{n-1}{n \choose n-j}5^j}{6^n-5^n}=\...
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Prove that if a is an infinite cardinal, then for each finite cardinal n, $a^n = a$

The proof in my textbook is the following: $a^n = card\:A^n$, the set of all maps of $I_n$ into a set $A$ of cardinal $a$. Since $A^{n} \subset I_n \times A$, $a^n \leq na = a$. On the other hand, for ...
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What exactly is being asked in Spivak's Calculus, Ch. 11, "Significance of the Derivative", problem 49, regarding Cauchy Mean Value Theorem?

The following is a problem from chapter 10, "Significance of the Derivative", from Spivak's Calculus: Prove that the conclusion of the Cauchy Mean Value Theorem can be written in the form $...
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How do formal proofs work and relate to interpretations?

As far as I know statements in formal logic are written in a (formal) alphabet which are just symbols, where the allowed sentences have to follow certain rules. If they do, they are called well formed....
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Doubt about a geometric solution to Problem $10$ of the $2017$ AIME I contest

The following is Problem $10$ of the $2017$ AIME I contest Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{...
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Chapter 11, Theorem 5.2 (4) of James Dugundji Topology

Let $p:X \to Y$ be a perfect map. Then: $(4)$ If $X$ is $2^\circ$ countable, so also is $Y$. Dugundji’s proof: Let $\{U_i\}$ be a countable basis for $X$, and let $\{V_i\}$ be the family of all ...
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Proof of small-span theorem in Apostol's Calculus Volume 1

The span of f is defined as follows: Let $f$ be real-valued and continuous on a closed interval $[a,b]$ and let $M(f)$ and $m(f)$ denote, the maximum and minimum values of $f$ on $[a,b]$. We shall ...
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Spivak Calculus: $f$ continuous on $[a,b]$, $n$-times differentiable on $(a,b)$, with $n+1$ roots in $(a,b)$. $f^{(n)}(x)=0$ for some $x$ in $(a,b)$.

The following is a problem from chapter 10, "Significance of the Derivative", in Spivak's Calculus Suppose that $f$ is continuous on $[a,b]$, that it is $n$-times differentiable on $(a,b)$, ...
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question on a proof Dirichlet's approximation theorem

We will fix some positive integer $N$ and consider the fractional parts of the numbers $0,α,2α,...,Nα$. Stick them into this collection of $N$ intervals $$[0;\frac{1}{N}),[\frac{1}{N},\frac{2}{N}),...,...
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Spivak Calculus: $f$ satisfies $f''(x)+f'(x)g(x)-f(x)=0$, for some g. Prove that if 𝑓 is 0 at two points, then 𝑓 is 0 on the interval between them.

The following is a problem from chapter 11, "Significance of the Derivative" from Spivak's Calculus Suppose that $f$ satisfies $$f''(x)+f'(x)g(x)-f(x)=0\tag{1}$$ for some function $g$. ...
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Showing that the intrinsic mean of continuously distributed data on the unit circle is almost surely unique

The article I am currently reading is Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 7. Below is a screenshot of the authors' proof that the intrinsic mean ...
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Figuring out a proof that $\mathbb{P}\{-\pi \leq X \leq -\pi + \delta\}=\frac{\delta}{2\pi}+\frac{\delta^{k+1}}{(k+1)!}f^{k}(-\pi+) + o(\delta^{k+1})$

The article I am currently reading is Intrinsic Means on the Circle: Uniqueness, Locus and Asymptotics by Hotz and Huckerman, pp. 4-5. Suppose that $X$ is a random variable living in the unit circle ...
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Husemoller: homotopy of linear clutching map (proposition $4.5$, pag. $187$)

Background : I'm currently studying vector bundle through the book of [husemoller,"fibre bundles"] (https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller). The following question concerns a ...
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-1 votes
2 answers
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How to prove $\frac{1}{\sqrt{a} + \sqrt{b}} = \sqrt{a} - \sqrt{b}$?

My daughter is learning how to rationalise surds for her school exams. One example being worked through is the following: $\frac{1}{\sqrt{6} + \sqrt{5}}$ In the tutorials she is following, the first ...
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Probability of alternating heads/tails - infinite coin tosses

We toss a fair coin until we get two consecutive heads or two consecutive tails. Each sequence of $n$ coin tosses has a probability of $\frac{1}{2^n}$. What is the probability that this experiment ...
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3 votes
2 answers
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Chapter 11, Theorem 5.2 (2) of James Dugundji Topology

Let $p:X \to Y$ be a perfect map. Then: $(2)$ If $X$ is regular, so also is $Y$. Dugundji’s proof: Given $y\in U$ in $Y$, there is by 1.5(b) an open $V \subseteq X$ with $p^{-1}(y) \subseteq V\...
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Clarification of Bondy's proof that an Ore graph on $n$ vertices has $\geq \frac{n^2}{4}$ edges

This question concerns a proof from Bondy, Pancyclic Graphs 1 of the fact that a graph $G$ on $n$ vertices satisfying the condition that $d(u) + d(v) \geq n$ for any non-adjacent vertices $u,v$ has at ...
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Existence of conformal equivalence between doubly connected domain and annulus

The following math overflow post https://mathoverflow.net/questions/261535/mapping-the-doubly-connected-domain-to-an-annulus provides a sketch proof of the fact that any (non-degenerate) doubly ...
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Triple bar meaning in proof of the Principle of Superposition

What does the triple bar mean in this context? "Thus $y(t)≡0,...$"
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1 answer
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Confused on two parts of Proof in C* algebras by Murphy

I'm working through the proof of the following theorem from C* algebra by Murphy. For context, $T_{\varphi}: H^2(T) \longrightarrow H^2(T)$ is given by $T_{\varphi}(f) = p(\varphi f)$ for $p: L^2(T) \...
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2 votes
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Proving that an associative binary operation gives rise to a group [duplicate]

I'm trying to prove the following claim. Let $S$ be a nonempty finite set, equipped with an associative operation $*: S \times S \to S$ such that, for every $x,y \in S$, there exists $z \in S$ such ...
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1 vote
1 answer
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Understanding a proof about $\lambda$-supercompact cardinal

I have trouble understanding the proof of this Lemma 20.15 from Jech's Set Theory, could someone explain why is $(2^\alpha)^M = (\alpha^+)^M = \alpha^+$?
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Artin, Chapter 2, Misc.6

I am trying to solve miscellaneous exercise 6 in Chapter 2 of Artin's book, Algebra. Below is the statement of the problem. Let $a = (a_1, \ldots, a_k)$ and $b = (b_1, \ldots, b_k)$ be points in $k$-...
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1 vote
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Hyperplane arrangement : The Shi arrangement

I have been lately reading Hyperplane arrangement lectures by Richard Stanley on https://www.cis.upenn.edu/~cis610/sp06stanley.pdf . In lecture 5, Theorem 5.16 we define the characteristic polynomial ...
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Please help me understand a proof for Taylor's Theorem

When using integration by parts, the proof appears to turn the latter function $(x-t)^k$ into the additive inverse of its integral $-(x-t)^{k+1}/(k+1)$. I don't understand how this could possibly be ...
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A sequence towards to the upper bound. [closed]

Let $A\subseteq\mathbb{R}$ be a no empty set with upper bound $m:=\sup{A}\in\mathbb{R}$, then it's simple to prove that exists a sequence $\{x_n\}\subseteq A$ such that $$\large x_n\to m$$ for $n\to \...
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Query on Serre's Proof

Hello I was reading Serre's book Linear Representations of finite groups and had a doubt while reading a proof of a proposition (Proposition 24, pg. 61): Proposition 24. Let A be a normal subgroup of ...
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Why is $\mathbf{u}=\alpha\ \mathbf{e}_k$?

I've read this answer to a question about Symmetric Rank Update (SR1). In this approach, we require the update of the Hessian matrix to be of the form $$ \mathbf{B}_{k+1}=\mathbf{B}_{k} + \mathbf{u}\...
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-1 votes
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Automata Regular Expression that remembers n iterations

Given is $L = \{\sigma_1 ~u~\sigma_2~v~\sigma_3 ~|~ \sigma_{1,2,3} \in \Sigma,~~ u,v\in \Sigma^*,~ |u|=|v|,~ \sigma_2=\sigma_3 ~or~ \sigma_2=\sigma_3 ~~\mathbb{but ~~ not ~~ both} \}$ I do not ...
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Prove $d(f,g)= \min {|f(x)-g(x)}|$ in $C[a,b]$ is not metric

Trying to answer whether we can set in the set of real continuous functions in the closed interval $[a,b]$ metric $d(f,g)$ I found this For the min case, let $a=−1$ and $b=1$ and consider the ...
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Confused on a set inclusion in C* algebras by Murphy

Im stuck on the following part of this theorem: The closed vector subspaces of $L^2(T)$ invariant for the bilateral shift $v=M_{z}$ (for $z: T \longrightarrow \mathbb{C}$ the inclusion map) are ...
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