For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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0
votes
2answers
26 views

Is $\sum_{n=1}^\infty a_n\sin(nx)$ converges on $[\varepsilon, 2\pi-\varepsilon]$?

Let $a_n$, a sequence monotonically decreasing to $0$. Consider $$\sum_{n=1}^\infty a_n\sin(nx)$$ Is the series converges uniformly on $[\varepsilon, 2\pi-\varepsilon]$? ($\varepsilon ...
4
votes
3answers
112 views

Is My Proof that $\pi^e < e^{\pi}$ Valid?

The other day, a math teacher at my college gave me a challenge problem: Prove that $$\pi^e < e^{\pi}$$ without using a calculator. The next day, I found a valid proof, but I used a log table ...
0
votes
1answer
24 views

Proof the statements

Proof the statements below i)If $P(A)=0$ and $B$ is any event, then $A$ and $B$ are independents ii)If $P(A)=1$ and $B$ is any event, then $A$ and $B$ are independents iii)The events ...
1
vote
1answer
16 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
2
votes
1answer
21 views

Are the running products of iid RVs independent?

Are the running products of iid RVs independent? Let $Y_0, Y_1, ...$ be independent random variables with $P(Y_n = 1) = P(Y_n = -1) = 1/2 \ \forall n = 0, 1, 2, ...$ (*) Define $X_n = Y_0 Y_1 ... ...
0
votes
1answer
30 views

Proof that any connected Graph has at least $n-1$ edges

I would really appreciate if someone could check this proof i though. Bare in mind i learned this subject in another language so i apologize in advance for my english. By Induction: $G$ connected ...
1
vote
1answer
35 views

Principle of well ordering

Every non-empty set $A\subset\mathbb{N}$ have a smallest element, i.e. an element $n_0\in A$ such that $n_0\leq n$ $\forall n\in\mathbb{A}$ Proof: Let $I_n=\{p\in\mathbb{N};p\leq n\}$ the set ...
0
votes
2answers
23 views

Lebesgue Integral of an Indicator Function using Measure Theory

Let $X$ be a random variable on $\Omega$ and fix $c \in \mathbb{R}$. I recently saw the following in a calculation: $$ \int_{\Omega} \mathbb{I}_{(c,\infty)}(X(\omega)) dP(\omega) = P(X \geq c). $$ I ...
2
votes
1answer
35 views

Subgroup proof verification.

Let $G$ be an abelian group, K is a fixed positive integer. $H$={$a\in$ $G$ $|$ $|a|$ divides K} . Prove that $H$ is a subgroup of $G$. My way of proving (Let me know how I could make it better or ...
2
votes
0answers
28 views

Help regarding a proof about Dedekind finiteness

I got this one as an exercise. If $F$ is Dedekind finite and $t \notin F$ then prove that $F \cup \{t\}$ is also Dedekind finite. I gave this as an answer: If $F\cup \{t\}$ is Dedekind infinite then ...
4
votes
1answer
37 views

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...
10
votes
1answer
84 views

Every function $f: \mathbb{N} \to \mathbb{R}$ is continuous?

This is a question that came up as a true false question in my textbook, and I was wondering what you thought of my reasoning. I claim that even though a graph of such a function doesn't look ...
1
vote
0answers
20 views

Rank of the sum of two rank 1 matrices, proof check

Claim: $(\forall u\in \mathbb{R}^2)$ $(\nexists(\delta,v)\in(\mathbb{R}, \mathbb{R}^2))$ such that $uu'+vv'=\delta \begin{pmatrix} 1 & 0\\0 & 0 \end{pmatrix}$. That is, for any vector $u$ of ...
0
votes
4answers
44 views

Logic, writing proof

i)Suppose that $x$ and $y$ are real numbers. Prove that if $x\neq 0$, then if $y=\frac{3x^2+2y}{x^2+2}$ then $y=3$ ii)Suppose that $x$ and $y$ are real numbers. Prove that if $x^2y=2x+y$, ...
1
vote
1answer
17 views

Question in proof from James Milne's Algebraic Number Theory

I'm having difficulty understanding a step in a proof from J.S. Milne's Algebraic Number Theory (link). Here $\zeta$ is a $p$th root of unity and $\mathfrak p = (1-\zeta^i)$ for any $1\leq i\leq p-1$ ...
1
vote
1answer
26 views

Explicitly decompose $\mathbb{C}^3$ into irreducible representations of $S_3$.

Consider the permutation representation of $S_3$ acting by permuting the elements of a basis of $\mathbb{C}^3$. Explicitly decompose $\mathbb{C}^3$ into irreducible representations. Can someone ...
2
votes
1answer
63 views

prove that this number contains two equal digits

We delete the first digit from the number $7^{1996}$ and then we add it to the remaining number, repeat this until we get a number consisting of $10$ digits, prove that, this number contains two equal ...
0
votes
2answers
24 views

Prove that for all $x$, $y$ in $\mathbb{R}$ there exist $z$, $g$ such that $x = z + g$, $y = z - g$

if I want to prove the following: $\forall x, y \in \mathbb{R}\,\,\,\exists\,\,z, g : x = z + g, y = z - g$ Can the resolution of the following system act as a proof: $\begin{cases} x = z + g\\ y ...
1
vote
0answers
14 views

Proposed proof for quasi-metric result

A quasi-metric on a set $X$ is mapping $\rho: X \times X \rightarrow [0, \infty)$ satisfying the following conditions: $\rho(x,y) \geq 0~~\text{and}~~\rho(x,x) = 0;$ $\rho(x,z) \leq \rho(x,y) + ...
3
votes
2answers
25 views

Check proof of some simple inequality

Can you check please my proof of this inequality? It's all right?
2
votes
0answers
48 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...
1
vote
0answers
24 views

G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
2
votes
1answer
35 views

Examples of irreducible representations

Which of the following representations are irreducible? 1) The tautological representation of $D_n$ on $\mathbb{R}^2$ 2) The action of $U(1)$ on $\mathbb{C}$ by multiplication 3) The ...
5
votes
3answers
77 views

Can a square be in the form $2x + 1$, when $x$ is odd?

I was given this question, and I think I have solved it, but I'm not sure it is correct because this differs from how the answer is given. What is the most common way to solve this problem? Let's ...
1
vote
2answers
20 views

Are the difference of two vectors orthogonal if the angle between the two vectors approaches 0? (Attempted proof)

Suppose that $\vec{a}=(x,y), \vec{a`}=(x', y'), \Delta \vec{a} = (x'-x, y'-y), \theta \rightarrow 0$ where $\theta$ is the angle between $\vec{a}$ and $\vec{a'},$ and the magnitudes are equal, $a=a'$ ...
1
vote
1answer
46 views

Find recursive formula - Question from exam, check my answer

We want to demolish and move a bridge from one location to another. The bridge is made out of $m$ road segments all connected $[0,1]$, $[1,2]$, $[2,3]$...$[m-1,m]$ We have a given function $f$ which ...
0
votes
0answers
87 views

sum and infinity

If you have the sums $ (1+2+..+n) + (1+2+3+..+n-1)+ (1+2+3+..+n-2)+(1+2+3+..+n-3)+...+(1+2+3)+(1+2)+1$for large enough $n$ $$\frac {n^3}{3!} \approx (1+2+..+n) + (1+2+3+..+n-1)+ ...
0
votes
2answers
42 views

Prove $f$ is Lipschitz on $K$

Let $f:\mathbb{R}^d\to \mathbb{R}$ such that it's partial derivatives are continuous. Let $K\subseteq \mathbb{R}^d$, a bounded set. Prove that $f$ is Lipschitz on $K$. My work: Since $f$'s ...
0
votes
1answer
27 views

Associated prime of $M/Q$ where $Q$ is $\mathfrak{p}$-primary

I need check if my statement is true and proof check (for some reason I couldn't find this anywhere): Let $Q$ be a $\mathfrak{p}$-primary submodule of $A$-module $M$. Then $\mathfrak{p}$ is the ...
0
votes
1answer
50 views

Prove the limit is $e^\alpha$

prove that $\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=e^\alpha$ $$\lim_{n \to \infty} \left(1+{\alpha\over n}\right)^n=\lim_{n \to \infty} \left(\left(1+{\alpha\over n}\right)^{n\cdot ...
3
votes
1answer
63 views

Proving a trigonometric identity with tangents [on hold]

Prove that: $$\tan^227^\circ +2 \tan27^\circ \tan36^\circ=1$$ any help, I appreciate it.
2
votes
1answer
29 views

Commutator ideal of reductive Lie algebra

I'm working through Fulton and Harris's book on Representation theory, and I've just done the exercise where I had to show: If $\mathfrak{g}$ is a reductive Lie algebra (defined as $Z(\mathfrak{g}) = ...
1
vote
3answers
65 views

Implies in a truth table, unclear. [duplicate]

In my textbook, we have the following truth table: $P$ true and $Q$ true means that "$P \implies Q$" is true. $P$ true and $Q$ false means that "$P \implies Q$" is false. $P$ false and $Q$ true ...
1
vote
1answer
30 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
-6
votes
0answers
36 views

Prove by induction this notation [closed]

Prove by induction? For $n\geq0$, $$\sum_{i = 0}^n (n i+2)^2={1 \over 3}(n+1)(2n+1)(2n+3)$$ Please help me.
1
vote
0answers
39 views

Verification of Basic Proof in Spivak Calculus (Induction)

I have began working through Spivak's Calculus book and trying to do the problems at the end of the chapters. I am rather new to proof, so forgive the naivety of this type of question. I am wanting ...
1
vote
0answers
41 views

Proof of Supporting Hyperplane Theorem from basic definitions.

My purposes in posting this question are twofold. First, I would like to have a lemma which I have proven on the way to proving the Supporting Hyperplane Theorem checked for rigor (zero tolerance for ...
0
votes
0answers
18 views

To circumscribe a square about a given circle.

http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV7.html I was just wondering something . We know that if a line touches a circle at one point, then this means that this line is forming a ...
0
votes
1answer
23 views

How many coins do we need to get $k$ amount

In the far away land of coinsville, they use $4$ different coins as currency, $\{1,10,100,200\}$ What is the computational class of the amount of coins (minimal!!) we need to get $k$ amount? Well, ...
3
votes
3answers
61 views

Strange integral test for convergence in my Analysis Script (proof flawed ?)

Today I was going through my Analysis Script which my Professor used for his course (meaning he often refers to it) and I found a Lemma called Integralcriteria for convergence of Series. I read its ...
2
votes
1answer
34 views

Please check my proof of this elementary covector result

I would appreciate it if someone could look over my proof and verify that it's correct. The question: Let $f$ be a $k$-covector on vector space V. Let $u_1,\dots u_k\in V$ and $v_1,\dots,v_k\in V$ ...
7
votes
5answers
961 views

Proof of the derivative of ln(x)

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) ...
1
vote
2answers
28 views

Prove that $H=V$ if $H$ is an $n$-dimensional subspace of an $n$-dimensional vector space $V$.

Prove that $H=V$ if $H$ is an $n$-dimensional subspace of an $n$-dimensional vector space $V$. I am not exactly sure what to do to show that $H=V$. So far I have reasoned that since $H$ and $V$ ...
2
votes
4answers
222 views

How do I know which of these are mathematical statements?

While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter. So how do I know if something is a mathematical ...
1
vote
1answer
36 views

Let $S$ be a subset of an $n$-dimensional vector space $V$, and suppose $S$ contains fewer than $n$ vectors. Explain why $S$ cannot span $V$

Let $S$ be a subset of an $n$-dimensional vector space $V$, and suppose $S$ contains fewer than $n$ vectors. Explain why $S$ cannot span $V$ Proof: Suppose $S$ is a subset of an $n$-dimensional ...
3
votes
2answers
140 views

Show that $f(x) = x^2$ is not uniformly continuous on $[0,\infty)$

Ok, I know the same question has already been asked here, and I am not looking for an answer even though my proof looks kind of the same. But, I need to know whether or not I am on the right track. ...
1
vote
2answers
86 views

if integral $f(x)\cdot g(x)=0$ mean that $f(x)=0$?

The question: If $f(x)$ is a continuous function, such that for every continuous function $g(x)$ defined over $[a,b]$ $$\int_a^b f(x)\cdot g(x)\,dx =0$$ does it mean that $f\equiv 0$? The ...
0
votes
1answer
31 views

The image of the inverse of a continuous function

First of all I'm not sure if my title is correct with the question, I find it hard to really get about what kind of set this question is about. It would be very helpful if someone could explain this ...
5
votes
1answer
166 views
+50

Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...
0
votes
0answers
19 views

finding the supremum

Let $A=\{x:\frac{[b\cdot n]}{n}\}$ when $n\in \mathbb{N}$ find the supremum we know that $b\cdot n-1<[b\cdot n]<b \cdot n$ therefore $b- \frac{1}{n}<\frac{[b\cdot n]}{n}<b$ so b is a ...