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

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0
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1answer
24 views

Legendre symbol, a theoretical question.

I need to show that if $p$ is a prime number of the form $p=4m+1$, then for any divisor $d$ of $m$: $$\left(\frac{d}{p} \right) = 1$$ where $\left(\frac{d}{p} \right)$ is the Legendre symbol. My ...
0
votes
1answer
30 views

Is the function continuous and differentiable at $x=-2$?

The function $f: (-3, \infty)$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; -3 < x < -2 \\ 1 & \mathrm{for} \; x = -2 \\ -x-e^{-x}+e^2-1 & \mathrm{for} ...
1
vote
1answer
27 views

Proof of (a step in the proof of) the Law of Large Numbers

Theorem: Let $f:[0,1] \to \mathbb R$ be a measurable function bounded by $c$. Let $U_1,U_2,\ldots,U_n$ be i.i.d. and Uniform$(0,1)$. Then: $$ P \left( \left\lvert ...
1
vote
1answer
23 views

In $\mathscr{V}$, let $X \subset \mathscr{V}$ be a set of $n$ vectors. $Y \subset X$ contains vectors all scalar multiples, $X$ linearly dependent.

I would just like to verify that my proofs are sound and receive any suggestions on rewording. (If relevant, I am self-studying and haven't done a serious proof in about a year.) $\mathscr{V}$ is a ...
0
votes
1answer
18 views

Prove $\vdash\forall x(\alpha\to\beta)\to(\exists x\alpha\to\exists x\beta)$

I want to show that: $\vdash\forall x(\alpha\to\beta)\to(\exists x\alpha\to\exists x\beta)$ I started my deduction as follows: $\vdash\forall x(\alpha\to\beta)\to(\forall ...
2
votes
2answers
52 views

$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
0
votes
0answers
24 views

Banach Tarski Notation

Okay, I think I have a full notation and the rules of it how to extend the Banach-Tarski Paradox to an abritary number of cutoffs, as I introduced in Another way of extending the Banach-Tarski ...
3
votes
2answers
61 views

My proof that an n digit number, times an n digit number can be expressed as a 2n digit number

I am very proud to say this is the first time I've actually used maths to endeavour to prove something without it being related to a question from my course! Statement In a base $B$, an $n$ digit ...
0
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2answers
28 views

Conditional expectation of random variable

I have this home assignment in Introduction to Probability, and I'm not comfortable with definitions and heuristics. I really need someone to check if I'm even in the right direction. The question: ...
1
vote
1answer
29 views

Prove a limsup and liminf inequality.

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence and let $B_n = \frac{1}{n} \sum_{i=1}^n a_i$ for each $n \in \mathbb{N}$. Prove that $\liminf a_n \le \liminf B_n \le \limsup B_n \le \limsup a_n$. ...
4
votes
2answers
37 views

How many ways to stack $n$ boxes of 4 colours so that no $2$ blue boxes are consecutive.

Let $X_n$ denote the number of ways to stack red, white and blue and green boxes, find the ways to count the ways of stacking n boxes, with no consecutive blue boxes. My attempt: Let $X^R_n$ denote ...
2
votes
1answer
29 views

Show that if $(X,d)$ is compact then, every open covering of $X$ has a Lebesgue number.

Let $(U_i)_{i \in I}$ be an open cover of a metric space $(X,d)$, a number $\epsilon >0$ is called a Lebesgue number of $(U_i)_{i \in I}$ if for all $x \in X$ exist $j \in I$ such that ...
0
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0answers
21 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
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0answers
15 views

discrete mathematics matrix relation proof [on hold]

Show that if MR is the matrix representing the relation R, then M[n] R is the matrix representing the relation Rn.
2
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2answers
31 views

How to establish the distributive property of sum notation

Establish the following property of sum notation: $$\sum_{i=1}^{n}(a_i+b_i) = \sum_{i=1}^{n}a_i + \sum_{i=1}^{n}b_i$$ I have tried in two ways. My first try uses recursive induction: ...
0
votes
0answers
19 views

What is a purely inseparable extension?

There are many different definitions of purely inseparable extension, and below is what I have chosen for my definition. (Since I don't know what is a standard one, if you know please tell me what ...
3
votes
1answer
61 views

Please check my demonstration of de l'hopital's rule

I have demostrate the de l'hopital theorem but in some steps I'm not 100% sure; The theorem I demostrate is for: $\lim_{x\rightarrow a+} \frac{f'(x)}{g'(x)}=L \implies\lim_{x\rightarrow a+} ...
1
vote
2answers
29 views

Prove trigonometric identity, hence or otherwise find the general solution

The following question requires one to prove the below trigonometric identity $$\cos 3x = 4\cos ^3 x - 3\cos x$$ Hence, or otherwise, find the general solution of the following equation $$(4\cos ^2 x ...
2
votes
1answer
26 views

Normal convergence: $\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$

I want to prove that: $$\sum_{n = 1}^{\infty} \frac{x}{(1 + (n-1)x)(1 + nx)}$$ is not normally convergent on $[a, \infty)$ for fixed $a>0$. Let $U_n(x)$ denote the general term. We have: ...
2
votes
1answer
38 views

Is my answer correct? (Devious auction game)

(Taken from here) The question was A man is auctioning a real $20\$$ bill. There are a vast number of bidders. A person may make as many bids as he wants. The starting bid is $5\$$. No $2$ ...
0
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0answers
25 views

Basis of square matrices

Find a basis of the space of complex $n \times n$ matrices, all the elements of which are invertible matrices. I suggest the following: using transvections for $i\neq j$ $T_{i,j}(1) := ...
0
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0answers
22 views

Triangles which are on the same base and in the same parallels equal one another.

I have a small question regarding proposition 37 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html The only problem I got with the proof is the fact that we ...
0
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0answers
23 views

associated prime of a module

Let $f: A\rightarrow B$ be a homomorphism of Noetherian rings, and $M$ a $B$-module. Question: Is $^af(Ass_B(M))=Ass_A(M)$? If $q$ is an associated prime of the $B$-module $M$, $p=^af(q)$, then from ...
1
vote
1answer
29 views

Proving limit of a 2D sequence using $\epsilon-N$ definition

I'm practicing proving a limit of a sequence $a_n:\mathbb{N}\to\mathbb{R}^2$ and I would like to know if I'm doing things correctly. The original limit is ...
0
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0answers
13 views

Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel

I have a small question regarding proposition 33 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI33.html We want to prove that two lines joining equal parallels ...
0
votes
1answer
15 views

If $G$ is a connected graph with $n$ vertices and$ n - 1$ edges then $G$ is a tree, using Induction.

I am still new to proof methods and not sure if this is the correct use of induction. Base case: $n = 1$ has $0$ edges and is a tree. Assume every connected graph with $k$ vertices and $k-1$ edges ...
3
votes
0answers
34 views

$\Box u= | u |^2 u$ global solution in $C^\infty$

Let $u_0, u_1 \in C^\infty( \mathbb{R},\mathbb{R}^3)$. Consider the cubic defocusing NLW $$(\ast)\begin{cases} \Box u= |u|^2 u \\ (u,\partial_t u) \restriction_{t=0} = (u_0,u_1) ,\end{cases}$$ where ...
0
votes
1answer
24 views

Polynomial irreducibiliy with substitution (need evaluation of logic)

One thing I have seen several times when trying to show that a polynomial $p(x)$ is irreducible over a field $F$ is that instead of showing that $p(x)$ is irreducible, I am supposed to show that $p(ax ...
2
votes
2answers
59 views

Proving $0x=0$ in a ring

I am trying to prove the above trivial statement. I am aware of the standard approach of letting $0 = 0 + 0$ and cancelling, but I would like the below statement to be verified/corrected: $1\cdot ...
1
vote
1answer
17 views

Can I assume a condition in the consequent?

Im reading Axler's Linear Algebra Done Right. In an exercise, he ask to prove that $$a\in F,v\in V,av=0 \implies a=0 \lor v=0 $$ where $V$ is a vector space over the field $F$. I've proved it this ...
0
votes
2answers
28 views

If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$

Let $f:(X,\tau_X)\to (Y,\tau_Y)$ Prove: If $\forall V\subseteq X$ where $x\in \overline V; f(x) \in \overline{f(V)}$, then $f$ is continous in $x$. Could someone verify the following proof? ...
0
votes
2answers
35 views

Showing there is a projection between a normed space and a subspace

Problem: Let $E$ be a normed space. Suppose $A$ is a finite dimensional subspace of $E$. Show that there exists a continuous projection $T: E \to A.$ Proof. I can write $E=A\oplus B$, where $B$'s ...
3
votes
0answers
41 views

Elemenatry topological proof of Erdos conjecture on Arithmetic Sequences

Define $C_n(A) = \{a \in A : \forall d \in \Bbb{N}$ one of $a + d, a +2d, \dots a + (n-1)d$ is not in $A\}$ For $n \leq m$, we have $C_n(A) \subset C_m(A)$. Proof. Let $x$ be in the LHS. Then ...
1
vote
2answers
71 views

For which $x \in \mathbb{R}$ does the series $\sum_{n=1}^\infty \frac{x^n}{n!}$ converge?

One problem of my exercise book asks for which $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \frac{x^n}{n!}.$$ The answer given by the exercise book is $|x|\leq 1$, but ...
1
vote
0answers
16 views

Continuity of Component Function

Let $f:Z\times X \to Y$ be given such that $f$ is continuous. I'm trying to prove that $f(z, -)$ is continuous for a fixed $z\in Z$. I would appreciate if someone could tell me if the proof that ...
5
votes
1answer
77 views

For what values of $x$ does the series $\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$ converge?

I have to study the values of $x$ for which $$\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$$ converges. First we say that we must have $x>0$. Then, I have started by rewriting the series as ...
0
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0answers
21 views

Counting the number of elements $x$ between $p$ and $p^2$ where lpf$(x(x+2))=7$

Let $p > 7$ be any prime. Let $f_7(p)$ be a function that counts the number of elements $x$ where $p < x < p^2$ and lpf$(x(x+2))=7$ where lpf is the least prime factor. It has been ...
3
votes
2answers
70 views

Theorem 7.2 in General Topology by S. Willard

Theorem 7.2 If $X$ and $Y$ are topological spaces and $f:X \to Y$ , then the following are all equivalent :- I) $f$ is continuous. II) for each E $\subset X$ , $f(\bar E) \subset ...
1
vote
1answer
16 views

Proving that in a complete graph $\lambda(K_{n}) = \delta(K_{n})$

Since $K_{n}$ is n-1 regular. Then $\lambda(G)$ must be n-1. Since $\lambda(K_{n}) \leq \delta(K_{n})$ then by definition they must be equivalent. Can I use the definition or should I say since ...
1
vote
1answer
17 views

Proving that in a complete graph $\kappa(K_{n}) = \delta(K_{n})$

Since $K_{n}$ is n-1 regular. Then $\delta(G)$ must be n-1. Since $\kappa(K_{n}) \leq \delta(K_{n})$ then by definition they must be equivalent. Am I approaching this proof the right way?
2
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0answers
55 views

Need help in understanding the question. Elemntary number theory

I have this question in my home assignment. I contains two parts and I don't quite understand what is the difference between them.The question is: Let $n > 2$ be an integer such that ...
-4
votes
2answers
51 views

Mathematical proof is correct?? [closed]

Edit2: I am giving you more information, basically I give the following derivation based on my understanding but my fellow saying it has problem that's why I want to confirm it from you guys. (1) ...
1
vote
1answer
45 views

Prove or disprove: If $a^2 \mid bc$, then $a \mid b$ or $a \mid c$

Prove or disprove: If $a^2 \mid bc$, then $a \mid b$ or $a \mid c$. I have not been able to find a counter example so I am thinking it may be true. I started by thinking that since $a^2 \mid bc$, ...
0
votes
1answer
36 views

If $D$ is a dense linear subspace of $X$ then $D\to Y$ extends to $X\to Y$ uniquely

I am trying to prove the following, but I am not confident in my work. Let $D$ be a linear subspace of a normed space $X$ that is dense in $X$. Let $Y$ be a Banach space. Show that any bounded ...
0
votes
2answers
25 views

Understanding Proof that $\mathbb{R} \setminus A$ is dense. Verify proof.

Here's the proof I was given but with two minor? differences Proposition.- If $A$ is countable then $\mathbb{R} \setminus A $ is dense. Proof: Suppose otherwise, then there exists real numbers $a$ ...
1
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1answer
35 views

Is this bullet really needed in Furstenberg's proof of infinitude of primes?

See here . The bullet I'm referring to is: Any union of open sets is open: for any collection of open sets $U_i$ and $x$ in their union $U$, any of the numbers $a_i$ for which $S(a_i, x) \subset ...
1
vote
1answer
22 views

Maximum $r$ such $p^r$ divides ${2n \choose n}$.

It is known that the number $n!$ contains the prime factor $p$ exactly $$ \displaystyle\sum_{k\geq1}\left\lfloor\dfrac{n}{p^k}\right\rfloor $$ Then, if for a fixed prime $p$, define $R(p,n)$ to ...
5
votes
3answers
49 views

If $a\in\mathbb{Q}$, prove that the sequence $\{\sin(n!a\pi)\}_{n=1}^\infty $ has a limit.

This exercise is from Methods of Real Analysis by Richard Goldberg. If $a\in\mathbb{Q}$, prove that the sequence $\{\sin(n!a\pi)\}_{n=1}^\infty$ has a limit. I think this proof relies on the ...
1
vote
1answer
20 views

Discrete metric, countable basis?

Give an example of a metric space which does not have a countable basis. I was thinking of some uncountable set, with a metric which results in an uncountable number of open subsets. Which ...
2
votes
0answers
60 views

Union of infinite broom and topologist's sine, connectednes, locally connectednes properties…

I'd like to know if my answer of the following exercise is correct. I really appreciate any suggestion you can provide to improve my argument or corrections in case I made a mistake :) Let ...