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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How is this step of simplification valid in a proof of martingale maximal inequality?

I'm reading a proof of $L^p$ maximal inequality from these notes. In the proof, we have $\left\|X_n^*\right\|_p^p \le C_p^p\left\|X_n\right\|_p \left\|X_n^*\right\|_p^{p / q}$ after Hölder's ...
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How the author uses MCT to complete his proof of Doob’s martingale maximal inequalities?

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $M=(M_t, t\ge 0)$ a continuous martingale with respect to a filtration $(\mathcal F_t, t\ge 0)$. Let $T>0$ and $p > 1$. Let $X := ...
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The measurability in a proof of Doob’s martingale maximal inequalities

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $M=(M_t, t\ge 0)$ a continuous martingale with respect to a filtration $(\mathcal F_t, t\ge 0)$. Let $T>0, \lambda >0$, and $p \...
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Understanding this proof about measurable function

I cannot understand this proof. Be $(X, \mathcal{F})$ a measurable space. If $f_n : X \to \bar{\mathbb{R}}$, with $n\in\mathbb{N}$ are measurable, then $\sup f_n$ is measurable. Proof $\forall a\in\...
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What's the thought process behind a mathematical proof? [closed]

How do mathematicians begin thinking about a mathematical proof? Mathematical papers make it appear as though the author has started at the axioms and built their way up to a theorem, almost as a side ...
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In Modal Logic, if something is true, is it necessarily true? $P\implies\square P$ [duplicate]

I'm new to modal logic and I am trying to understand it more intuitively. If something is true, is it necessarilly true? I.e. $$P\implies\square P$$ This seems intuitive but it is not an axiom. This ...
1 vote
1 answer
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Clarification of the details of the proof of Cayley Hamilton theorem in commutative algebra

I am trying to understand this proof of the Cayley Hamilton theorem from commutative algebra by Atiyah Mcdonald. So I am reading the following power point slides which gives more details but there is ...
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Proof by contraposition. Let $x$ be an integer. If $8$ does not divide $x^2-1$, then $x$ is even.

Prove by contraposition: Let $x$ be an integer. If $8$ does not divide $x^2-1$, then $x$ is even. Assume $x$ is odd. Prove $8|x^2-1$ So, $x=2n+1$ for some integer $n$. Then $x^2-1=4(n^2+n)$ for some $...
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$\lnot A \lor \lnot B$ ∴ $\lnot(A \land B)$ TFL help [closed]

This question is in my textbook and I have tried making some attempts but there is no solution manual. I tried following the first rule for negation but after that I am completely lost. I have lost ...
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1 answer
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Elementary explanation of getting two consecutive $6$'s in a die roll experiment

I know that there are already numerous questions that adress this problem. However, I am not interested in a soltuion at all but in an explanation of a particular solution (see https://math....
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1 answer
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If $x\in \Bbb{Q}$ then $x^2=2$ [closed]

If $x ∈ Q$ and $x = \sup\{q ∈ Q | q > 0, q^2 < 2\}$ then $x > 0$ and $x^2 = 2$. Proof: Let $E$ equal the set on the right hand side, and suppose $x \in Q$ such that $x = \sup E$. Then, since $...
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Is a proof "The diagonals of a parallelogram bisect each other" without using concept of "congruence of angles" still correct?

I post my proof in pictures below and I am not sure if my proof correct? This is an exercise from section 1.1 of the classical textbook linear algebra by Stephen Friedberg etc Prove that the ...
3 votes
3 answers
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Understanding this passage in Borel Cantelli Lemma N.2

I'm trying to understand a passage in the proof of Borel Cantelli Lemma 2. Be $(\Omega, \mathcal{F}, P)$ a probability space and $(A_n)$ a sequence in $\mathcal{F}$ such that $(A_n)$ are pair ...
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Prove that if a normal subgroup of $A_n$ contains even a single $3-cycle$ it must be all of $A_n(n\geq 5)$

Prove that if a normal subgroup of $A_n$ contains even a single $3-cycle$ it must be all of $A_n(n\geq 5)$ The solution given is as follows: Let $\pi=(i_1 i_2 i_3)\in S_n$ be a $3-cycle.$ Then, for ...
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Prove that if a normal subgroup of $A_n$ contains even a single $3-cycle$ it must be all of $A_n.$ [closed]

Prove that if a normal subgroup of $A_n$ contains even a single $3-cycle$ it must be all of $A_n.$ The solution given is as follows: It's good to start from $(1 2 3), (a b c)$ where $a, b, c$ are ...
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Why is there no T in the simplified derivative?

Why is the expression I have circled lacking the variable T? from simplifying the expression to the right of it, it seems T should be multiplied with the numerator. For context, I have attached the ...
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How does $\frac{y'}{x^2}-\frac{2y}{x^3}=0$ imply $\frac{d(x^{-2}y)}{dx}=0$? [closed]

On the Wikipedia page for Integrating Factors, https://en.wikipedia.org/wiki/Integrating_factor, they mention in an example that the equation, $$\frac{y'}{x^2}-\frac{2y}{x^3}=0$$ Can be written as, $$\...
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Find the locus of the point of intersection of three mutually perpendicular tangent planes to the paraboloid $ax^2+by^2=2cz.$

Find the locus of the point of intersection of three mutually perpendicular tangent planes to the paraboloid $ax^2+by^2=2cz.$ The solution given in a regional handout(or extract) is hereby attached: ...
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Is assumption of drawing uniform points on a random circle valid in described scenario

When a golfer wants to make his putt, he needs to estimate the slope of the green / surface. Suppose a golfer estimates the green slope to be $$ \theta=(\theta_x, \theta_y) $$ where $θ_x$ and $θ_y$ ...
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A few $(3)$ questions regarding Spivak's proof of the Inverse Function Theorem.

I have a few questions regarding Spivak's proof of the Inverse Function Theorem: Theorem: Let $f$ be a function $\mathbb{R}^n\to\mathbb{R}^n$. If $\ \ \ \ a)$ $f$ is $C^1$ in an open set containing $...
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Problem in understanding the result of finding the equation of a tangent plane to a central conicoid.

I was studying about conicoids. There was a topic about finding the tangent plane to a conicoid (general). It went on like this: Let the equation of the conicoid be $ax^2+by^2+cz^2=1.$ Any line ...
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is the definition of the complex exponential function arbitrary? [duplicate]

If I cite my textbook the complex exponential function is defined as: $$ e^{\theta i} = cos \theta + i \sin \theta $$ Is this just an arbitrary definition or is it possible to prove this somehow?
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Limit points of a subset are limit points of the superset - confusion

Need some help in understanding this SE post. The proof given by OP seems valid. Yet, top-rated answer provides a more complex proof, that I don't understand. I don't understand what is the problem ...
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2 votes
2 answers
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lists with length $k$ with elements $b_1,b_2,\dots,b_k$ such that $|b_1|+|b_2|+···+|b_k| \le n$

Ivan and Alexander write lists of integers. Ivan writes all the lists of length $n$ with elements $a_1,a_2,\dots,a_n$ such that $|a_1| + |a_2|+\dots+|a_n| \le k$. Alexander writes all the lists with ...
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1 vote
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show that $\lim\limits_{n\to \infty}\sup (-1)^{n}n=\infty$

How do i show formally that : $\lim\limits_{n\to \infty}\sup (-1)^{n}n=\infty$. I know that if n is odd, then lim inf will be $-\infty$ and if n is even then lim sup will be $\infty$. However, i dont ...
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Question about derivation ($\lambda$-abstraction)

How exactly is line 17 obtained using $abst$? The $abst$ rule is stated here (where $s\in \{\ast, \square\} $), and one of its premises is that $\Gamma \vdash \Pi x : A. B $ but there aren't any $\Pi$...
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Proof for Generating function of Bessel function

I have already seen the relative questions Generating function for Bessel function Prove the generating function for the Bessel function. but I need some help for the reasoning behind a specific ...
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Three Questions about Convex neighborhoods lemma 4.1 in Riemannian Geometry (Do Carmo)

Proof of Lemma 4.1 I have two questions about the proof of Lemma 4.1. I think the blue line should be the closure of geodesic ball, $\overline{B_r(p)} \subset W$. Because $q$ lies in the boundary of $...
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I didn't understand a detail from the Schrijver et al.'s proof of the Tutte-Berge formula

I read the proof of the Tutte-Berge formula from the book "Fundamentals of Graph Theory" by Allan Bickle (page 204), but one detail I did not understand. For a graph to have a 1-factor, ...
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If $f(X) = X^m + a_1X^{m-1} + \ldots + a_m = \prod_{i=1}^m (X - \alpha_i)$, then $|\alpha_i| \le \max\{1,mB\}$ where $B = \max|a_i|$

In Remark $1.17$ on Pg. $13$ of Milne's Fields and Galois Theory, the following observation is used to describe an algorithm for factoring monic polynomials in $\Bbb Z[X]$ (more generally, polynomials ...
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In the group $(\Bbb Z, +)$, if $H$ is the smallest subgroup containing $4$ and $6$, then which one of the following is true? [closed]

In the group $(\Bbb Z, +)$, if $H$ is the smallest subgroup containing $4$ and $6$, then which one of the following is true? (i)$H=24\Bbb Z,$ (ii) $H=12\Bbb Z,$ (iv)$H=2\Bbb Z,$ (v)$H=\Bbb Z$ The ...
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Proving $(\forall a,b \in \Bbb R)(a<b\implies(\exists r\in\mathbb Q)(a^7<r-4<b^7))$

Prove the following statement: $$(\forall a,b \in \Bbb R)(a<b\implies(\exists r\in\mathbb Q)(a^7<r-4<b^7))$$ Hint: You may need to use the theorem: Given any two real numbers $a < b$, ...
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Consecutive Integer Sum of an m-member Sequence is Divisible by m Proof

This problem is found in Richard A. Brauldi's book on Introductory Combinatorics. It goes as follows: Given m integers $a_1, a_2, ... ,a_m$, there exist integers $k$ and $l$ with $0 \le k \lt l \le m$ ...
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Define X=R\{k} and define ⋆ to be the operation such that x⋆y=(x−k)(y−k)+k. Does this satisfy closure?

Define X=R{k} and define ⋆ to be the operation such that x⋆y=(x−k)(y−k)+k. Check each of the four axioms of a group (closure, associative, identity, inverse). Which of them hold? Is (X,⋆) a group. It ...
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Proof clarification of hausdorff measure equals lebesgue measure in one dimension

In the book measure theory and fine properties of functions of Evans and Gariepy, i'm trying to understand the proof of theorem 2.2 (ii) which states that $\mathcal{L}^1 = \mathcal{H}^1$. Here the ...
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Atiyah-MacDonald 5.16: Geometric interpretation of Noether normalization lemma

I am working on the second part of Exercise 5.16 of Atiyah-MacDonald, which involves proving that over an algebraically closed (and thus infinite) field $k$, given an affine algebraic variety $X\in k^...
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Let $\sim$ be some equivalence relation on $X$. Is there some function with domain $X$ such that $f(x)=f(y)$ exactly when $x\sim y$?

1. Suppose that $X$ is a set. a) Let $f$ be some function with domain $X$ (and codomain anything you like), and say that $x \sim y$ means $f(x)=f(y)$. Is $\sim$ an equivalence relation? If it is, then ...
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1 answer
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Uniformly distributed measures are unique up to multiplicative constant.

I am reading the book Geometry of sets and measures in euclidean spaces of Mattila and Im having trouble understanding the last part of the proof of theorem 3.4 Definition: Let $X$ be a metric space ...
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2 answers
53 views

Help: induction exercise $\sum_{i=1}^{n}\left ( \frac{1}{i^{2}} +i\right )\leq \frac{n^{3}+n^{2}+4n-2}{2n}$ [duplicate]

I' m doing this induction exercise: $\sum_{i=1}^{n}\left ( \frac{1}{i^{2}} +i\right )\leq \frac{n^{3}+n^{2}+4n-2}{2n}$ where $n\geq 1$ I ve proved step p(1) , now i m doing the induction step. I ve ...
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Formal proof about why $\operatorname{E}[(X-\operatorname{E}[X|Y])^2|Y]$ is minimal

Reading different books of mathematical statistics (that I remember now: Lehmann and Casella in their point estimation book, Batthacharya et al. in their mathematical statistics book, and probably ...
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How can I convince myself that the steps in euclid's algoritm are valid?

I'm banging my head against Euclid's algorithm at the moment and I think I need some external input in order to gain a breakthrough.... Let's say we have a fraction like: $$ \frac{216}{66}=3*66+18 $$ ...
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Show that $S:=\{x \in \Bbb R: 1 \leq x^3 \leq 2 \}$ is nonempty and bounded above. Let $a = \sup S$. Show $1 \leq a \leq 1.5$ and $a^ 3 = 2$

Define $S:=\{x \in \Bbb R \mid 1 \leq x^3 \leq 2 \}$. a)Show that $S$ is nonempty and bounded above. b)Let $a$ be the supremum of $S$. Show that $1 \leq a \leq 1.5$. c)Show that $a^3 = 2$. [You may ...
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Proving that a topological space is not metrizable

Given $X = \{p,q\}$ with topology given by $\emptyset, \{p\}, X$, I am trying to prove that $X$ is not metrizable. The lecture notes that I am working through do not introduce the Hausdorff property ...
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All models of the successor function

I'm having problems understanding proofs to similar questions so I want to check if my understanding is correct. The version I have uses three axioms: $s$ is injective Everything except $0$ is in the ...
2 votes
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Exercise D2, Cook and Nguyen, Logical Foundations of Proof Complexity

Cook and Nguyen, Logical Foundations of Proof Complexity, on p. 42, ask for a proof in $I \Delta_0$ of $\exists z(x + z = y \lor y + z = x)$, and give as hint, "Induction on $x$. Base case: B2, ...
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Ahlfors Riemann sphere to stereographic projection

In the chapter on complex numbers, Ahlfors makes a link between associating a point on the sphere to stereographic projection. However I'm unsure of the details of how he does this. He writes that ...
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Rudin's proof for Riemann Integrability of Composition of functions

I'm having some trouble with a certain step for the proof of the following theorem Suppose $f\in\mathcal{R}(\alpha)$ on $[a,b]$, $m\le f\le M$, $\phi$ is continuous on $[m,M]$, and $h(x)=\phi(f(x))$ ...
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Show almost sure convergence of sequence of random variables

Let be $(X_n)_{n\in\mathbb{N}}$ a sequence of random variables. Show that $P\left(\bigcap\limits_{m=1}^{\infty}\bigcup\limits_{n=1}^{\infty}\bigcap\limits_{k=n}^{\infty}\left\{\left|X_k-X\right|<\...
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Theorem 10, Section 6.7 of Hoffman’s Linear Algebra

Theorem 9: $V=W_1\oplus …\oplus W_k$$\iff$$\exists E_1,…,E_k\in L(V,V)$ such that (i) each $E_i$ is projection ($E_i^2=E_i$) (ii) $E_iE_j=0$, if $i\neq j$ (iii) $I=E_1+…+E_k$ (iv) $R_{E_i}=W_i$. ...
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Why does this prove that $a^n$ goes towards infinity?

My textbook is using some kind of strange technique using binomials to prove a point. Since they're using this techniques on several other occasions I need to understand what they are doing and why. ...
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