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

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1
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2answers
32 views

Squaring both sides of an inequality: attempt to prove a general rule

I have attempted to produce a proof of the intuitive rule for squaring inequalities, according to which, given any two numbers x and y and regardless of their sign, 1) if |x| < |y| then ...
0
votes
0answers
37 views

Topology on partially ordered set

Let $(X, \leq )$ be a partially ordered set. How would you define a topology on $X$ such that the closed sets are precisely the order-closed sets? Where $B \subset X$ is order-closed if ...
0
votes
0answers
15 views

Question concerning a proof on Stroock and Varadhan 1971

In the proof of theorem 2.3 of the article diffusion processes with boundary conditions (1971) one reads: where $Q_{s,x}$ is the unique solution to the martingale problem for $a,b$ starting from $x$ ...
2
votes
0answers
47 views

Proof check:$ \left | \mathbb{R} \right |= 2^{\left|\mathbb{N} \right |}$

This is my first time to post here. Sorry if this post is too simple or naive. Here I would like to prove that $\left | \mathbb{R} \right |= 2^{\left |\mathbb{N} \right |}$ I would first ...
1
vote
0answers
16 views

A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
1
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3answers
37 views

Showing that a subgroup of an abelian group is normal—is this sufficient?

When asked to show that a subgroup $H$ of the abelian group $G$ is normal, does it suffice to say: first, $H$ is a subgroup, so it contains the identity element of $G$ and inverses $h^\prime$ for ...
0
votes
2answers
17 views

Replacing occurrences of the same integer as follows: is it legitimate in subsequent steps of a proof?

I need to spot the incorrect step in a spoof (false proof) and the first two lines are: $-5 = -5$ $-7+2 = -4-1$ Of course both the right- and left-hand side of the equation in the second line are ...
2
votes
1answer
47 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
1
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1answer
38 views

How many surjective functions $f: X \to \{1,…,j\}$?

How many surjective functions $f: X \to \{1,...,j\}, |X|=j \cdot k.$ can be defined if they must satisfy: $$ |\{x\in X: f(x)=r\}|=|\{x\in X: f(x)=s\} \forall r,s\in \{1,...,j\} $$ My attempt: From ...
-1
votes
0answers
65 views

sum and infinity, agen

I posed this question a few days ago that is no answer, but now I will clarify. 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 ...
1
vote
1answer
50 views

Is my proof of closedness of multiplication operator corect?

I am considering an operator $A: L^2(\mathbb R , d \mu) \supset D(A)\to L^2 (\mathbb R, d\mu)$ defined by $(Af)(x)=a(x)f(x)$ for known measurable function $a$. Domain is of course all those functions ...
3
votes
4answers
103 views

Taking limits on each term in inequality invalid?

So this inequality came up in a proof I was going through. $$c - 1/n < f(x_n) \leq c$$ Where $c$ is a real number, $f(x_n)$ is the image sequence of some arbitrary sequence being passed through a ...
3
votes
5answers
156 views

Is :$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$ irrational or transcendental or real number?

Is there someone who can show me if :$$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$$ is irrational or real or transcendental number ? Thank you for any help
1
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2answers
46 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
4answers
171 views

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

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
30 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
26 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
32 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
41 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
37 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
25 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
40 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
32 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
39 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
106 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
21 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
47 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
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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
27 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
64 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
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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
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0answers
16 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
28 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
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0answers
25 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
36 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
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1answer
47 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
92 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
43 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
28 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
51 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 [closed]

Prove that: $$\tan^227^\circ +2 \tan27^\circ \tan36^\circ=1$$ any help, I appreciate it.
2
votes
1answer
30 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 ...
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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
40 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 ...