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

learn more… | top users | synonyms

1
vote
0answers
62 views

IMO 1983 Solution - Day 1 Problem 3

The questions goes as follows: Let $a$ , $b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc - ab - bc - ca$ is the largest integer which cannot ...
1
vote
1answer
32 views

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$.

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$. My Attempt: $(\rightarrow):$ $$\alpha i=ai-b-ck+dk \Rightarrow -b+ai+dj-ck$$ ...
2
votes
0answers
58 views

Non-Noetherian subring of F[X,Y]

I am trying to prove that, for a given field $F$, the subring $$R:=\{p(X,Y)=\sum c_{ij}X^iY^j \in F[X,Y] : c_{0j}=c_{j0}=0 \text{ whenever } j>0\}$$ of $F[X,Y]$ is not Noetherian. I think I ...
0
votes
1answer
44 views

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$.

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$. First, $\mathbb Q(z)\subseteq \mathbb Q(z^3,z^5)$ would be trivial, right? Then we ...
2
votes
2answers
25 views

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$.

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$. I understand that $\mathbb Q(\sqrt{a})$ is the smallest ...
2
votes
1answer
81 views

Prove that $\mathbb{Q}\!\smallsetminus\!\mathbb{Z}$ is dense in $\mathbb{R}$

Can someone just tell me if this is a correct way to prove it. let $(a,b)$ be a nonempty open interval in $\mathbb{R}$. Then by density of $\mathbb{Q}$ in $\mathbb{R}$ there exists $q\in \mathbb{Q}$ ...
3
votes
1answer
60 views

Proposed limit proofs

I have a question regarding an approach to finding the limit of a function: Consider the function $$f(x) = x^{2}\sin(\frac{1}{x})~~~\text{for }x \neq 0~~~\text{and }f(0) =1$$ Show that ...
1
vote
2answers
25 views

Find a $\delta_1$ and a $\delta_2$

I have the majority done on this problem but the final step I'm having trouble understanding how to prove. I have put my proof so far in the answers. Can someone look at it and tell me if I'm showing ...
-1
votes
1answer
45 views

Legendre's Conjecture: Bounded Prime Gaps

I have encountered some error in the details of what Legendre's conjecture implies about bounded prime gaps. So I am working to correct errors and to state both what is conjectured and what is implied ...
1
vote
1answer
39 views

Proving a Special Case of a Limit Theorem

I'm having trouble proving a special case of the limit theorem below. I attempted a proof by contradiction that appears to me to make sense in the first direction but I'm not able to come up with ...
0
votes
2answers
33 views

Is this proof that $f(x) \in X$ valid?

I just finished this question from my exercises for Foundations and Proof. The problem states; Let $ X = \{x \in \mathbb{R}: x \neq 1\} $. Define $ f:X \to X$ by $f(x) = \frac{x+1}{x-1}$ For $x \in ...
1
vote
2answers
37 views

Determinant n exponent

Let $A$ be an $n\times n$ matrix. Prove that $\det A^n=(\det A)^n$. Proof by induction. Suppose $n=1.$ $\det A^n= (\det A)^n$ $\det A^1=(\det A)^1$ $\det A= \det A$ Now assume $n=k$ is true. ...
0
votes
1answer
18 views

Need correction for my “proofs” about the integrable functions.

Let $(X,\mathcal{A},\mu)$ be a measureable space, and assume that $\mu(X)<\infty$. Let $\left \{ u_{n} \right \}_{n\geq 1}$ be a sequence of functions in $\mathcal{L}^{1}(\mu)$ that converges ...
1
vote
1answer
46 views

Is a group structure is need for this analysis

We know that $ℤ/nℤ$ is a finite set, then it is possible to find a bijection $$θ:ℤ/nℤ→T={1,2,...,n}$$ where $T$ is a finite part of natural numbers ℕ. Let us consider the set $G=ℤ^{r}×ℤ/nℤ$, ...
4
votes
1answer
36 views

A positive integer $n$ is prime iff $\varphi(n)! \equiv -1 \pmod n$

Is this proof acceptable ? Theorem 1 (Wilson) A positive integer $n$ is prime iff $(n-1)! \equiv -1 \pmod n$ Theorem 2 A positive integer $n$ is prime iff $\varphi(n)! \equiv -1 \pmod n$ . ...
1
vote
0answers
31 views

Correct process for proof in graph theory.

I'm working on what I'm sure is a fairly basic proof in graph theory. I must prove that $Every$ $graph$ $G$ $contains$ $a$ $path$ $with$ $\delta(G)$ $edges$. $\delta(G)$ is the minimum degree of the ...
18
votes
5answers
414 views

Find the value of $\sqrt{10\sqrt{10\sqrt{10…}}}$

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
0
votes
1answer
95 views

Gaps between primes: bounds - a question of possibilty [closed]

Let $n$ be any given natural number. Let $p$ be the very next prime greater than $n$. Let $b$ be the bound for the prime gap above $n$. Here, the bound is strictly the limit from $n$ to $p$, meaning ...
0
votes
0answers
17 views

Polynomial Expansion Proof Estimation

1-(1-x)^n where x is a value between 0 and 1 and n is a large value. This estimates to around x*n. I am having trouble with the polynomial expansion. According to Pascal's triangle, the first few ...
0
votes
2answers
23 views

Simplification of a function

I have to simplify the following function : $g(x)$ = ${\sin x\over \sin 1} \cos (x-1)-{\sin x\over \sin 1} \cos 1 + {\sin (x-1)\over \sin 1} \cos 1-{\sin (x-1) \over \sin 1} \cos x$ My attempt: => ...
2
votes
0answers
42 views

Proving a PDE has a particular weak form (check my proof please!)

Let $u_t - \Delta u = f$ hold in $L^2(0,T;H^{-1})$ for a solution $u \in L^2(0,T;H^1_0)$ with $u_t \in L^2(0,T;H^{-1})$. This means $$\int_0^T \left(\langle u_t(t), v(t)\rangle + \int_\Omega \nabla ...
0
votes
3answers
50 views

Is my arithmetical proof using induction correct?

The exercise 2.b of my textbook ask me to prove that: $$\text{(P): }\;\forall n\in \mathbb{N}, 13\;|\;(3^{n+2}+4^{2\cdot n+1})$$ I would like to know if my proof is correct and if not what I need to ...
0
votes
1answer
17 views

Analysis: Proof checking and help on 2nd part (Integrals)

So I have the question, $f(x)= x$ if $0$ $\leq$ $x$ $\leq$ $1$ and $f(x)= x+2$ if $1<x$ $\leq$ $2$ (the same f(x) I just couldn't figure out how to do the big bracket) Part 1 is asking me to ...
-1
votes
2answers
32 views

Verify the Identity [duplicate]

$\binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k}$ So far I have gotten $\frac{n!}{(k-1)!\big(n-(k-1)\big)!} + \frac{n!}{ k! (n-k)!}$ But I quickly lose myself once I have to start making the ...
0
votes
4answers
38 views

Verifying my proof that if $|S(x)| \leq 1$, then $\lim_{x \to 0} x\cdot S(1/x) = 0$

Question: Suppose that $S : R \to R$ is a function so that for all $x$, $−1 \leq S(x) \leq 1$. Prove from the limit definition that $$\lim_{x\to0} x \cdot S(1/x) = 0.$$ This is my ...
2
votes
1answer
19 views

Epsilon-N method - Proof verification

Prove (using the epsilon-N method) that the sequence of numbers $\dfrac{5n^3-2}{n^3}$ converges. Calculate the limit first. First we calculate the limit: $\lim_{n \to \infty} \dfrac{5n^3-2}{n^3} ...
0
votes
1answer
38 views

Verification of convolution between gaussian and uniform distributions

Let $n \sim \mathcal{N}(\mu, \sigma^2)$ and let $u \sim \mathcal{U}(a,b)$, with $b>a>0$, and suppose that $n$ and $u$ are independent random variables. Let $g = n + u$. The probability density ...
3
votes
2answers
50 views

$\varepsilon-\delta$ proof of $\lim_{x \to -\infty} \frac{1}{1+x}=0$

I do not have a clue about where to start. If I'm right, I need to find a relation between $\varepsilon$ and $\delta$ such that $0<|x + \infty|<\delta$ implies $|\frac{1}{1+x}|<\varepsilon$. ...
2
votes
1answer
34 views

Limit of Inner products.

I had the following appear on an exam, and I can't see why I'm wrong and have no clear explanation from the professor. My answer is below. Let$ X$ be a Hilbert space, and let $\mathcal{E}$ be an ...
-1
votes
1answer
47 views

Prove a=v*dv/dx

Using calculus, and assuming a particle moving along the x-axis is concerned, prove that $a=v*dv/dx$ ~~~~~~~~~~~~ this is what I did, but im not sure it's rigorous enough: $a=dv/dt$ $t=x/v$ ...
2
votes
0answers
38 views

If $G_1\cong G_2$ and $H_1\cong H_2$ then $G_1 \times H_1 \cong G_2 \times H_2$

If $G_1\cong G_2$ and $H_1\cong H_2$ then $G_1 \times H_1 \cong G_2 \times H_2$ Proof: $f_G:G_1\rightarrow G_2$ and $f_H:H_1\rightarrow H_2$. Question 1: Is the following statement valid? Does ...
0
votes
1answer
39 views

¬p ⊬ ⎕(p → q): Where's the mistake in my proof?

My professor noted on one of his slides that ¬p ⊬ ⎕(p → q). Intuitively, this seems correct; however, I can only prove that it is false. I suspect I've made a mistake in my proof. Where have I gone ...
3
votes
3answers
84 views

Is my proof by contradiction about the empty set correct?

I am trying to learn about proofs and one of the exercice in my book (Maths ABC) is about proof by contradiction. I think I understand the concept but I would like to have a feedback on the following ...
0
votes
0answers
8 views

Proving properties of nth roots

First let me define some things. Let $x \gt 0$ and $n \ge 1$. Now $x^{\frac{1}{n}}:=\sup\ [ y\in \mathbb R : y \ge 0 \text{ and } y^n \le x]$ (a) If $x \gt 1$ then $x^{\frac{1}{k}}$ is a decreasing ...
0
votes
0answers
6 views

Proof: Monotonicity of heuristic

I'm trying to proof the consistency of a heuristic for a problem, namely the chebyshev distance: The heuristic is given by: $$H(N,G) = \max(|N_x-G_x|, |N_y - G_y|)$$ And represents the estimated ...
1
vote
0answers
53 views

Prove that ordinal multiplication is left distributive

Suppose $\alpha, \beta$ and $\gamma$ are ordinals. Prove the distributive law $\alpha \cdot ( \beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$. The following is my proof: Proof: We use ...
0
votes
1answer
38 views

Prove that the following function tends to infinity as z approaches i (complex analysis)?

Prove: $\lim_{z\to i} [2 /({1 + z^2})] = \infty$ My attempt: We want $M > 0$ such that if $0 < |z - i| < \delta$ then $|2/(1 + z^2)]>M$ Now $|2/(1 + z^2)|>M$ ...
2
votes
0answers
47 views

Every graph G contains a path with d(G) edges (proof critique).

Curious to see whether my proof below is acceptable or not. Any feedback will be well received. Many thanks. Every graph G contains a path with $\delta(G)$ edges. $\mathbf{Proof.}$ Let $G$ be a ...
-1
votes
2answers
153 views

A New, Possible Proof of the Infinitude of the Primes?

$$1=1$$ $$2=2$$ $$3=3$$ $4=2\cdot2$ At $4$, the first prime number, $2$, is there as a factor. So I say that at the square of $2$, $2$ comes into play as a prime factor. At this point, $2$ is the ...
1
vote
1answer
36 views

Show that $\frac{\mathrm{d^{2}}B }{\mathrm{d} A^{2}}> 0 $ if $U''<0$.

Given, $A = W_0 - L_0 + I - qI$, $B = W_0 - qI$, and $EU = p U(A) + (1-p) U(B) = k$, where $k$ is a constant. $\frac{\mathrm{d} B}{\mathrm{d} A}\bigg|_{}^{EU=k} = \frac{\frac{\partial EU }{\partial ...
1
vote
4answers
13 views

Differentiability question ends up in contradiction.

Let $f(x)=x^3cos\frac{1}{x}$ when $x\neq0$ and $f(0)=0$. Is $f(x)$ differentiable at $x=0$? My first attempt Definition: A function is differentiable at $a$ if $f'(a)$ exists. $$f'(x)=\lim_{h ...
0
votes
1answer
28 views

transitivity of subformula relation - proof

Problem: prove that the relation "is a subformula of" is transitive for propositional formulas. let $\phi \in Sub(\psi)$ and $\psi \in Sub(\chi)$ prove that $\phi \in Sub(\chi)$. my proof: if $\phi ...
0
votes
1answer
25 views

Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
2
votes
2answers
246 views

Is it true that $3$ is the only prime of the form $n^2-1$? [closed]

One less than a perfect square is prime if and only if the prime is 3. Is this really, really true and do we have proof?
1
vote
0answers
25 views

Check my proof - Linear Algebra

Still not completely confident with my capabilities in writing formal proofs so I thought I would ask for a check of this proof. Theorem Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be ...
2
votes
3answers
90 views

What is wrong with this calculation of $\binom{\frac{1}{2}}{k}$?

The reason I ask this question is that I want to show that: \begin{equation*} \binom{\frac{1}{2}}{k} = -2\frac{1}{k}\binom{2k-2}{k-1}\left(-\frac{1}{4} \right)^k \end{equation*} \begin{align*} ...
0
votes
1answer
19 views

To Prove That Field of Fractions of Given Rings is Same. Proof Verification.

I am trying to solve Q. 8a in Section 9.1 from Abstract Algebra by Dummit & Foote. The problem is: Let $F$ be a field and $R=F[x,x^2y,x^3y^2,...,x^ny^{n-1},...]$ be a subring of $F[x,y]$. ...
1
vote
1answer
25 views

Show that $\lim_{n\rightarrow \infty } Var(Y_{n}) = 0$.

Given $Var(Y_{n}) = (\theta_{2}-\theta_{1})^2\frac{n}{(n+2)(n+1)^2}$. My work: $$lim_{n\rightarrow \infty } Var(Y_{n}) = \lim_{n\rightarrow \infty } (\theta_{2}-\theta_{1})^2\frac{n}{(n+2)(n+1)^2} = ...
0
votes
1answer
57 views

Equicontinuity if the sequence of derivatives is uniformly bounded.

I would really appreciate if someone could look over this proof for me. Let $ \left\{ g_m \right\} $ be a sequence of functions defined on an interval $ [a,b] \subset \mathbb{R}^n$. Let $ \left\{ ...
0
votes
2answers
56 views

Find the largest natural number m such that n$^3$-n is divisible by m for all n$\in$ $\mathbb{N}$.

Find the largest natural number m such that n$^3$-n is divisible by m for all n$\in$ $\mathbb{N}$. Prove your assertion. So my basis that I have is: Notice that (1)$^3$-(1)=0, and m(0)=0, so m ...