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Questions tagged [proof-verification]

For questions concerning a specific proof or a specific solution, asking for verification, identifying errors, suggestions for improvement, etc. (You should not use this tag if the question does not contain a proposed proof/solution.)

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Can I split this inequality like this?

Recently I had solved this number theory problem but after I solved it I was a bit uncertain whether my approach was correct so I approached AOPS. The problem is : Prove that $[x] + [y] + [x + y] \...
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Riemann integrability of Thomae function on $[0,1]$

Here is my proof using Riemann sum. We know that by Archimedian property, for any $\epsilon>0$, there exists $n\in\mathbb{N}$ such that $\frac{1}{n}<\frac{\epsilon}{2}$. Let $A_n = \{x|g(x)\...
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Proof Verification Short Exact Sequence Rank Theorem

I am trying to prove the rank-nullity theorem for short exact sequences; if $R$ is an integral domain, and $M',\,M,\,M''$ are all $R$-modules with $$0\rightarrow M' \xrightarrow{\psi} M\xrightarrow{\...
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1answer
18 views

Integral test with $f$ negative and increasing?

The Integral Test states Assume $f$ is continuous, positive, and decreasing on [$1, \infty$). If $\int_1 ^{\infty}f(x)\,dx$ exists and is finite, then $\sum f(n)$ converges and vice versa. ...
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27 views

Jech Set Theory (3rd Edition) Exercise 7.16

This question is about exercise 7.16 from Jech's Set Theory (3rd Edition): Let $\kappa$ be a regular uncountable cardinal, and let $A$ be a set of size at least $\kappa$. With $S \triangleq P_\kappa(...
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1answer
243 views

Fourier Transform of Airy Equation

I am trying to find $Y(k)$ of the equation $y''(x)-xy(x)=0$ and hence show that $$y(x)=\sqrt{\frac{2}{\pi}}\int_0^{\infty}\cos\left(\frac{k^3}{3}+kx\right) \ dk,$$ given $Y(0)=1$. Here, we use the ...
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1answer
58 views

Let $S=\{x~|~x\in G$ and $x^2 \in H\}$. Show that $S$ is a subgroup of $G$ for $H<G$, $G$ abelian.

The question states: Let $G$ be an Abelian group with subgroup $H < G$. Let $S=\{x~|~x\in G$ and $x^2 \in H\}$. Show that $S$ is a subgroup of $G$. My proof is different than what is in the ...
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Determining whether counting events is a renewal process

$\begin{array}{l}{\text { Consider a renewal process with mean interarrival time } \mu . \text { Suppose that each }} \\ {\text { event of this process is independently "counted" with probability } p ....
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How to use Wald's equation to determine expectation in gambling model?

$\begin{array}{l}{\text { In each game played one is equally likely to either win or lose 1. Let } S \text { be your }} \\ {\text { cumulative winnings if you use the strategy that quits playing if ...
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1answer
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If $x_n \rightarrow x \implies \lim_{n \rightarrow +\infty}$ $f_n (x_n) \rightarrow f(x)$ ,then $f_n \rightarrow f$ uniformly in each compact set $K $

I found this proof online and was having trouble understanding the last steps. Lemma: Given that $f_n(x_n)$ converges to $f(x)$ , show that $f_n$ is uniformly convergent to $f$ on a compact set k. ...
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1answer
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Show that $\varphi: \overline{E} \times K \longrightarrow N$ is continuous

Let $K$ compact in a metric space $M$, $N$ is a metric space, $\mathcal{C}(K,N)$ the space of continuous functions $f: K \longrightarrow N$ and $E \subset \mathcal{C}(K,N)$ a family of functions such ...
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2answers
31 views

Proof that every common divisor divides GCD (solve only by Bézout's identity)

As part of the course's assignments, we received a task to prove the following sentence using only Bézout identity: Every common divisor of $a, b$ divides the gcd $(a, b)$. I tried the following ...
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Given the multiplication morphism on the scheme functor, how to get it as scheme morphism?

I am studying group schemes. I would say I understand the definition of an $S$-group scheme as to give an $S$-scheme together with morphisms for multiplication law, identity, and inverse and also the ...
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1answer
27 views

A linear transform mapping one billinear map to another may not exist (counter-example)

$$\newcommand{\im}{\mathrm{im}\;}\newcommand{\Span}{\mathrm{Span}}\newcommand{\rank}{\mathrm{rank}\;}$$For a bilinear map $\phi : V \times W \to E$ define the first nullspace as $$ N_1(\phi) = \{ v \...
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1answer
37 views

If $f:[0,1]\to \mathbb{C} $ be continuous with $f(0)=0$ and $f(1)=2$, then $|f(t_0)|=1$ for some $t_0 \in [0,1]$

Question: Let $T=\{z\in \mathbb{C}:|z|=1\}$ and $f:[0,1] \to \mathbb{C}$ be continuous with $f(0)=0$, $f(1)=2$. Show that there exists at least one $t_0$ in $[0,1]$ such that $f(t_0)$ is in $T$. ...
3
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1answer
33 views

If $R$ is a right-Noetherian ring and the Jacobson radical satisfies the right Artin-Rees property, then $\bigcap^{\infty}_{n=1}\text{Jac}(R)^n = 0$

A (two-sided) ideal $I$ of a ring with identity $R$ has the right Artin-Rees property if for any right-ideal $E$ of $R$, there exists an integer $n\geq1$ such that $E\cap I^n\subseteq EI$. If $R$ ...
2
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1answer
43 views

Homeomorphism between upper closed hemisphere and disk.

Let $H^2$ be the closed upper hemisphere, that is $$H^2=\big\{(x,y,z\in\mathbb{R}^3)\;|\;x^2+y^2+z^2=1, z\ge0\big\},$$ and let $D^2$ be the closed unit disk $$D^2=\big\{(x,y)\in\mathbb{R}^2)\;|\;x^2+y^...
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0answers
31 views

For an increasing function $f$, if $f(y_n)-f(x_n) \to 0$ , then $f$ is continuous at $0$

Question: Let $f: \mathbb{R} \to \mathbb{R} $ be an increasing function. Suppose there are sequences $(x_n)$ and $(y_n)$ such that $x_n<0<y_n$ for all $n\geq 1$ and $f(y_n)-f(x_n) \to 0$ as $n \...
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0answers
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prove or disprove:$f'(x)$ can not be injective over $(a,b)$.

Problem Prove or disprove the following statement: Let $f(x)$ be differentiable over $[a,b]$, and $f'(x)$ be continuous over $(a,b)$. $f'(a)=f'(b)=0$. Then there exist $x_1,x_2$ satisfying $a<x_2&...
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0answers
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How to Evaluate $\int_{0}^{\infty}\frac{x^2+2}{x^4+4} \ dx$ given $\mathcal{F}_c(e^{-x}\cos(x))=\sqrt{\frac{2}{\pi}}\frac{k^2+2}{k^4+4}$

I have previously shown that ($\mathcal{F}_c(f(x))$ denotes the Fourier cosine transform of $f(x)$) $$\mathcal{F}_c(e^{-x}\cos(x))=\sqrt{\frac{2}{\pi}}\frac{k^2+2}{k^4+4} \tag{1}.$$ Using this ...
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Prove that $f(x,0)=f(0,x)$ for all X (Hint:Use A1) Let f(x):{ f(x,0) if x≠0, Let f(x):{ 1 if x=0 [on hold]

Consider throwing a dart at the origin of the Cartesian plane. You are aiming at the origin, but random errors in your throw will producr varying results. We assume that: 1.) The errors do not depend ...
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1answer
40 views

Prove that $ \left \lfloor {\log n} \right \rfloor = \left \lfloor {\log \left \lceil \frac {n-1}{2}\right \rceil} \right \rfloor + 1$

We have been doing algorithm analysis in university, and after analyzing binary search algorithm, the following equation resulted. What we have to do now is to prove that $ \left \lfloor {\log n} \...
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1answer
22 views

Proof Verification: Show that every compact metrizable space has a countable basis

It would be appreciated if someone could review my proof for accuracy. Thanks! Show that every compact metrizable space has a countable basis Proof: Let X be a compact metrizable space. Then ...
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28 views

Composition operator and topological spaces

I am trying to complete this topological question I have a solution already I just would like for someone to check if for me. Question: My solution:
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23 views

restriction operators and continuous maps from Hom(X,Y) to Hom(A,Y)

I am trying to solve this topological question and would like to know if I am on the right track with my solution. As in if I have the correct answer or if I need to add or delete anything. However, I ...
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1answer
32 views

Show that the Compact open topology on Hom(X,Y) is hausdorff

I am trying to complete this topological question and I would like to know if my solution is correct. Any help would be greatly appreciated! My Solution: Let X be a topological space, and Y a ...
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Prove the orthogonal complement of a nonempty subset of a Hilbert space is closed.

Statetment Let $M \subset H$ be nonempty, and $H$ is a Hilbert space. Then $M^\perp$ is a subspace, and $M^\perp $ is closed. Proof To show that $M^\perp$ is a subspace we let $x,y \in M^\perp$, $a,...
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0answers
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Slightly alternative proof to the converse part of Cauchy's General Principle

I want to prove that: If $\forall \epsilon >0$, $\exists k \in \mathbb{N}$, such that $| u_{n+p}-u_n| <\epsilon $, whenever $n\geq k$, $p\in \mathbb{N}$, then $\{u_n\}$ is convergent. Proof: [...
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0answers
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Quotient of Module over PID

Let $R$ be a PID Let $a$ be a nonzero element in $R$ Let $M=R/(a)$ For any p of R prove that $p^{k-1}M/p^kM\cong R/(p)$ if $k\leq n$ $p^{k-1}M/p^kM\cong 0$ if $k> n$ where $n$ is the power ...
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1answer
24 views

Proving almost uniform convergence of $ n\sin(\frac{x}{n}) $

I'm learning about uniform convergence. For example, consider a function sequence $$f_n: \mathbb{R} \rightarrow \mathbb{R}, f_n(x) = n\sin(\frac{x}{n})$$ This sequence converges pointwise to $ f(x)...
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2answers
68 views

Prove by mathematical induction $4^n > n+1$

Prove the following by mathematical induction: $4^n > n+1$, for all integers $n ≥ 1$ Step 1: $n=1$: LHS $= 4^{(1)} = 4 $ RHS $= (1) + 1 = 2$ LHS > RHS. ∴ $P(1)$ is true. Step 2: Assume $P(k)$...
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5answers
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What's wrong with my proof that $f(x) = x^2, x^4, \ldots$ are bijective?

Let $n$ be an odd positive integer. Prove $f : \mathbb{R} \rightarrow > \mathbb{R}$ defined by $f(x) = x^n$ is bijective. My attempt: First we prove injectivity. Suppose we have $f(x) = f(y)$ so ...
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1answer
81 views

Prove that $\lim_{x\to\infty}\sum_{n=1}^{\infty}\frac x{n^2+x^2}$ exists and is positive

Show That $$\sum_{n=1}^\infty{1\over x ^2+n^2} \sim \frac1x$$ as $x\to \infty.$ It is enough to show that $\lim_{x\to\infty}\sum_{n=1}^{\infty}\frac x{n^2+x^2}$ exists and is positive $$=\lim_{x\to\...
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1answer
44 views

Counting integers with a least prime factor greater than $x$ in a sequence of $x$ consecutive integers.

It is well known from Sylvester-Schur that in any sequence of $x$ consecutive integers, there is always at least one integer divisible by a prime greater than $x$. I am interested in counting the ...
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1answer
29 views

Check on a proof I saw on another thread: Metrizable Lindelöf spaces have a countable basis

I saw the following proof given of to the theorem below. I don't think the proof is correct, but I wasn't quite sure as it was given an up vote and thought I'd re post here to get some other opinions. ...
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0answers
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Positive Homogeneity - proof

Consider the following risk measure: $\rho_{1-\alpha}(X) = inf_{z>0}\{z^-1(ln(\frac{E[e^{zX}]}{\alpha}))\}$, with $\alpha \, \epsilon ]0,1]$ Any insight in how to show that $\rho_{1-\alpha}(\...
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1answer
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Infimum of Function

How can I show that $$\inf_{z>0}\{z^{-1}(z\mu+0.5z^2\sigma^2-\ln(\alpha)\} = \mu+\sigma\sqrt{-2\ln(\alpha)}\quad ?$$ $\alpha \in\ ]0,1]$ Think I'm probably doing something wrong when computing ...
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1answer
29 views

Proof verification of the Open Mapping Theorem

In Hahn and Epstein's book "Classical Complex Analysis" book they proved the following Open Mapping Theorem as a consequence of the Rouche's theorem; for $f(z)$ and $g(z)$ analytic on and inside a ...
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Solutions to cubic and higher degree functions

If $u = ax^2+a_2x+a_3$ is a quadratic polynomial then there exists a solution to $w=v^2$ where $w=cu+c_2$ and $v = bx+b_2$, which can be easily shown to be true. For instance when $u=x^2+x+1$, $w=4u-...
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2answers
37 views

Uniform Convergence and continuity (Rudin 7.4)

This question has been asked for sequences, but I couldn't find it for series. Suppose $f(x) = \Sigma_{n=1} ^ \infty \frac{1}{1+n^2x}.$ $f(x)$ converges on $(0,\infty)$ and on $(-\infty, 0)$ ...
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0answers
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Find a basis for the row space, column space, kernel, and image of the following matrix verification

For the following matrix: $$ \begin{bmatrix} 1 & 2 & 1 & 3 \\ 2 & 5 & 5 & 6 \\ 3 & 7 & 6 & 11 \\ 1 & 5 & 10 & 8 \\ \end{bmatrix}...
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0answers
73 views

pointwise vs uniform convergence (Baby Rudin 7.4)

This is a basic question about the relationship between pointwise and uniform convergence. Suppose $f(x) = \Sigma_{n=1} ^ \infty \frac{1}{1+n^2x}.$ The question (Rudin 7.4) is what intervals does ...
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1answer
27 views

Showing that the simple continued fraction of $\sqrt{d}$ has period length 1 iff $d=a^2+1$

Given that I know if $d$ is an integer that $\sqrt{d}=[\alpha_0,\bar{\alpha_1},...\bar{\alpha_n},\bar{2\alpha_0}]$. I want to show that $\sqrt{d}$ has period length 1 if and only if $d=a^2+1$, for ...
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0answers
12 views

About the relationship between $f,g$ and its solutions for ODE

This is a true or false question. Let $f(x,t)$ continuous and $\dot{x}=f(t,x)$ an ODE with unicity of solution. So, given $\epsilon >0$, exists $\delta >0$ such that $$|g(x,t)-f(x,t)|<\...
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1answer
17 views

Formally deriving a vacuous truth from a definition involving conjoined implications

Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Longrightarrow |x| = x) \wedge (x < 0 \Longrightarrow |x| = -x)$ I want to use this definition in one of my proofs. So I ...
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3answers
27 views

Question on the injectivity of a function.

Consider the map $f\colon\mathbb{S}^1\to\mathbb{S}^1$ defined as $f(z)=z^2$, where $\mathbb{S}^1$ is the unit circle, $$\mathbb{S}^1=\{z\in\mathbb{C}:|z|=1\}.$$ On $\mathbb{S}^1$ we define the ...
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2answers
34 views

Proving an inequality involving absolute value; how do I justify using a conjunction (and) instead of a disjunction (or)?

I'm putting together the following the proof, and I have a question about one of the final steps. Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Rightarrow |x| = x) \...
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0answers
34 views

$f_n\to f$ in $L^1(\mu)\implies f_n $ are uniformly integrable

If $f_n\to f$ in $L^1(\mu)\implies f_n $ are uniformly integrable where $\mu $ is positive measure My Attempt: Uniformly Integrable family: $\{f_n\}_{n\in A}$ is said to be uniformly integrable if $...
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5answers
53 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
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1answer
37 views

proof that $Y$ follows normal distribution

I'm new to probability and studying multivariate normal distribution. The one thing I don't understand is the linear transformation of multivariate normal distribution.If the $X$ follows Normal ...