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

learn more… | top users | synonyms

0
votes
0answers
9 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 ...
2
votes
1answer
50 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
23 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
37 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
votes
0answers
22 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
votes
0answers
20 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
votes
0answers
18 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
28 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
votes
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
31 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
23 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
56 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
32 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
40 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
votes
0answers
20 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
64 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
votes
0answers
44 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 ...
0
votes
0answers
27 views

Checking the proof: Ric=Kg

Let $(M,g)$ be a semi-riemaniann surface (dim M =2). Let Ric be the Ricci tensor and K the sectional curvature. Then Ric = Kg. Proof: Let $R$ be the Riemann curvature of $(M,g)$. Let $x, y \in T_pM$ ...
-4
votes
2answers
51 views

Mathematical proof is correct?? [on hold]

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
35 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
24 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
vote
1answer
34 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
48 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 ...
2
votes
1answer
28 views

Where do I use the fact that $V$ is closed in $X$ in the following proof.

Let $V$ be a supspace of $(X,\tau)$ and $A\subseteq V$. Let $V$ be closed in $X$. Then $A$ is closed in $V$ if and only if $A$ is closed in $X$. Where the subspace $V$ has the relative ...
2
votes
1answer
36 views

Proving $R(3,4)\le 9$

I am trying to prove $R(3,4)\le 9$. This is my approach: For any $K_9$ we have (WLOG) at least 4 red edges by the pigeonhole principle. Consider all of the edges between these 4 red edges, if ...
1
vote
0answers
16 views

Show that $\{x\in V| \langle x,e \rangle=0 \forall e\in E\} =\{y\in V ~| ~y\perp w_i, 1\leq i \leq k \}$

Let $E$ be subset of a vector space $V$. Let $B =\{w_1,\dots,w_k\}$ be a basis for $E$. Prove: $E^\perp =\{y\in V | y\perp w_i, 1\leq i \leq k \}$ Is my proof correct? Define two sets: (a) ...
1
vote
0answers
17 views

Proving that a linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column.

Is this proof sufficient? Theorem: A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column. Proof: (Showing the implication $\to$) Suppose a ...
0
votes
2answers
58 views

$\mathbb{R}$ is uncountable with Cantors Diagonal argument (how to improve binary expansion specificity?)

I know it's spelled out more than usual, but this is an introduction to higher math class. If there's any way I can improve this, please let me know. Thank you in advance. Let ...
2
votes
1answer
25 views

Proving a collection of subsets is a basis

I am given this definition of a basis: Let $a$ be a point in a metric space $X$. A collection, $\mathfrak{B}_a$, of neighborhoods of $a$ is called a basis for the neighborhood system at $a$ if every ...
1
vote
4answers
31 views

Epsilon-Delta Continuity proof (verification/help)

So, I am really bad at these problems, and I don't know why. Edit: The metric over $\Bbb R$ is assumed to be $|f(a,b)-f(x_1,x_2)|$ Problem statement: Define $f: \Bbb R^2 \rightarrow \Bbb R$ by ...
-4
votes
0answers
39 views

Why is this clause in unsatisfiable? [closed]

Why is the following clause unsatisfiable? By unsatisfiable, I mean that we can derive to the empty clause using the Resolution Proof Method. $\{x\}, \{\lnot x\}, \{x, y, z, w\}$
1
vote
0answers
35 views

Proof verification: Bolzano-Weierstrass (second proof).

I construct an other proof of Bolzano-Weierstrass theorem (i.e. that all bounded sequence has a subsequence that converge). Do you think that my construction is correct ? Let $a:=\inf_{n\geq 1} x_n$ ...
1
vote
1answer
29 views

Proof Verification: Cauchy Sequences are convergent.

Assume $(a_n)$ is a Cauchy sequence, then it is bounded. By Bolzanno-Weierstrass theorem, there is a convergent sub-sequence $(a_{n_j})$; denote its limit as $a$. Thus we have the following: for every ...
1
vote
0answers
20 views

A sequence with two distinct limits

I just wanted to check I was right about this: Consider $X=\{1,2,3\}$ equipped with the topology $T=\{\emptyset,\{1,2\},X\}.$ Then the sequence $(1,2,1,2,1,2,\ldots)$ converges to both $1$ and $2$ ...
0
votes
0answers
43 views

Looking for a function which can serve as an upper bound to a count of the the pairs (x)(x+2) that have a given least prime factor?

Let $p \ge 7$ be a prime. Let $z > p$ also be a prime. Let $f_p(z)$ be the number of elements $x$ such that $z \le x < z^2$ and the least prime factor of $x(x+2) = p$ I am trying to find ...
2
votes
0answers
58 views

Definite Integral of Series

I'm looking to find someone to either verify or correct my computation of the definite integral $$\int_{0}^{1} f(x)~dx$$ where $$f(x) = \sum_{n=1}^{\infty} \frac{1}{(x+n)^{n}}$$ What I came up ...
3
votes
3answers
110 views

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. My attempt: Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$ As $f\mid g, g$ all the roots of $f$ are roots of $g$. This means I ...