If you already have a proof for some result, but want to ask for a different proof (using different methods).

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2
votes
1answer
51 views

If $|\nabla F| > 1$ and $|F| \le 1$, is there a zero nearby?

I saw this claim, stated without much explanation, in an article I'm reading: Let $F:\mathbb{R}^n\to\mathbb{R}$ be a $C^1$ function which satisfies $|\nabla F|>1$ everywhere. We know that ...
1
vote
1answer
58 views

Prove that $ \lim (s_n t_n) =0$ given $\vert t_n \vert \leq M $ and $ \lim (s_n) = 0$

Let $ (t_n) $ be a bounded sequence, i.e., there exists $ M $ such that $ \vert t_n \vert \leq M $ for all $ n $, and let $ (s_n) $ be a sequence such that $ \lim s_n = 0 $. Prove $ \lim (s_n t_n) ...
1
vote
0answers
73 views

Sequential version of the Eberlein-Shmul'yan theorem

Theorem: A Banach space is $(i)$ reflexive iff $(ii)$ every bounded sequence possesses a weakly convergent subsequence; see e.g. Thm 3.18 and 3.19 in Brezis' 2010 book. The implication $(i) \implies ...
2
votes
4answers
151 views

Prove that $ \lim_{n \rightarrow \infty } \frac{n+6}{n^2-6} = 0 $.

My attempt: We prove that $ \lim\limits_{n \rightarrow \infty } \dfrac{n+6}{n^2-6} =0$. It is sufficient to show that for any $ \epsilon \in\textbf{R}^+ $, there exists an $ K \in \textbf{R}$ such ...
1
vote
2answers
134 views

How many digits do we need for a proof ??

In the question: Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$, the value of that integral was ...
4
votes
3answers
198 views

Prove that $ \displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4} $.

My attempt: We prove that $$\displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4} $$ It is sufficient to show that for an arbitrary real number ...
3
votes
2answers
72 views

Simpler proof of $g\,h\,g^{-1} = h^a \Rightarrow g^n\,h\,g^{-n} = h^{a^n}$

In a rather easy online lecture on group theory (which included many obvious statements such as "the only divisors of a prime number $p$ are $1$ and $p$"), the professor began a proof by assuming that ...
4
votes
3answers
230 views

Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

My attempt: Consider the polynomial $ (x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0 $. By the rational zeros theorem, we can conclude that $ \pm 1$ and $ \pm 23 $ are the only possible rational solutions*. ...
1
vote
4answers
117 views

Prove Inequality without induction.

I showed this inequality by induction. I want other methods to prove it. $(\frac{2n}{3}+\frac{1}{3})\sqrt{n}\leq \sum_{k=1}^{n}\sqrt{k}\leq (\frac{2n}{3}+\frac{1}{2})\sqrt{n}$ Thank
5
votes
0answers
163 views

Elementary proof of irreducibility criterion

From ``Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented: Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If ...
2
votes
0answers
81 views

Sign of some permutation.

I trying to find the sign of this permutation: $\left(\begin{array}{cccccccccc} 1 & 2 & 3 & \cdots & \cdots & \cdots & \cdots & \cdots & n-1 & n\\ 2 & 4 & ...
3
votes
1answer
100 views

Proving that $(\cos \theta )^p\leq (\cos p\, \theta )$ for $0\leq\theta\leq \pi/2$ and $0<p<1$ through an alternative method?

I'm reading the Berkeley Problems in Mathematics book: Prove that $(\cos \theta )^p\leq (\cos p\, \theta )$ for $0\leq\theta\leq \pi/2$ and $0<p<1.$ I could find other ways to prove it, ...
2
votes
1answer
33 views

Looking for simpler proof of “well-definedness” of basis subtended by points in “general position”

Throughout this question, the variable $I$ represents the set of integers $\{0,\dots,p\}$, with $p > 0$. I'll use the ad hoc notation $I_j$ to denote the set $I \, \backslash \{j\}$, and ...
2
votes
0answers
100 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
2
votes
2answers
172 views

Is $f_n(x)=n^2x(1-x^2)^n$ uniformly convergent on $[0,1]$?

Does $f_n(x)=n^2x(1-x^2)^n$ converges uniformly on $[0,1]$ $\lim_{n\to \infty} f_n(x) =f(x)=0$ and then I calculated sup of $|f_n(x)-f(x)|$ which came out to be $\frac{n^2}{\sqrt{2n+1}}\cdot ...
12
votes
3answers
290 views

Simplified form for $\frac{\operatorname d^n}{\operatorname dx^n}\left(\frac{x}{e^x-1}\right)$?

I have found the following formula: $$\frac{\operatorname d^n}{\operatorname ...
4
votes
1answer
84 views

Simple method for $\frac{(2n+1)!}{(n!)^{2}}$ divide $lcm(1,2,\ldots,2n+1)$

The question is to prove that $\frac{(2n+1)!}{(n!)^{2}}$ divides $lcm(1,2,\ldots,2n+1)$. This seems like it should be a simple question, but try as I might, I can't seems to find any way that does ...
1
vote
0answers
47 views

An identity involving partial derivatives

Suppose $F(x,y)$ is a function of two variables satisfying $F(0,0)=0$. By differentiating some expressions, I obtained the identity $$ \frac{ \partial F}{\partial x}(x_0, y_0) = \int_0^1 ...
3
votes
1answer
75 views

Simple proof of some formula for n!

I have found an interesting identity for n! , but my proof is slightly complicated using Bernoulli numbers. Can somebody find some simple proof of the following formula? $$(-1)^n ...
3
votes
3answers
453 views

Prove $1 + \cot^2\theta = \csc^2\theta$

Prove the following identity: $$1 + \cot^2\theta = \csc^2\theta$$ This question is asked because I am curious to know the different ways of proving this identity depending on different ...
2
votes
4answers
210 views

Prove $1 + \tan^2\theta = \sec^2\theta$

Prove the following trigonometric identity: $$1 + \tan^2\theta = \sec^2\theta$$ I'm curious to know of the different ways of proving this depending on different characterizations of tangent and ...
0
votes
1answer
96 views

Is there a simpler, more abstract proof of the Cayley-Hamilton theorem for matrices?

The Cayley-Hamilton theorem is equivalent to: Let $R$ be a ring and let $M_n(R)$ be $n\times n$ matrices over $R$. Then the minimal polynomial of $A \in M_n(R)$ over $R$ divides the characteristic ...
5
votes
1answer
61 views

Sum of squares of cotangents (Check properly of expression)

I found exercise in "Introduction to algebra" Part I (A.I. Kostrikin) Check expression $\sum_{k=1}^n\cot^2\frac{k\pi}{2n+1}=\frac{n(2n-1)}{3}$ for $n=1,2,3,4,5$. For $n=1,2$ it is simple. ...
5
votes
2answers
224 views

Is there a proof of the irrationality of $\sqrt{2}$ that involves modular arithmetic?

I was reading Ian Stewart's Concepts of Modern Mathematics. Using congruences, It's possible to explain why all perfect squares end in $0,1,4,5,6,9$ but not in $2,3,7,8$. With this I had the ...
0
votes
1answer
66 views

A nice group isomorphism

Show that $$k(\mathbb{Z}/n\mathbb{Z})\cong (\gcd(n,k)\mathbb{Z})/n\mathbb{Z}.$$ I want to see as many as possible proofs of this nice fact.
2
votes
1answer
111 views

Proof Without Words for $GCD(a,b) \cdot LCM(a,b)=ab$

Is there any proof without words for the identity $GCD (a,b) \cdot LCM(a,b)=ab$ ?
1
vote
1answer
43 views

Integration involving non-elementary functions

Let \begin{equation}g(t)=\begin{cases}\frac{\sin{\frac{1}{2}}t}{t}, & t \not =0 \\ \frac{1}{2}, &t=0 \end{cases} \end{equation} Calculate $\text{lim}_{m\to ...
3
votes
1answer
366 views

Integrating Dirichlet's Kernel

Determine $\frac{1}{\pi}\int_{-\pi}^{\pi}\left[D_{m}(t)\right]^{2}dt$ for $m=100$ where $D_{m}(t)=\frac{1}{2}+\sum_{n=1}^{m}\cos{nt}$ (Dirichlet's kernel). Initially, I thought of using the ...
0
votes
0answers
45 views

elegant proof of Radon–Nikodym theorem

Do you know about an elegant proof of Radon–Nikodym theorem, which is not as cumbersome as the usual ones?
1
vote
1answer
81 views

How we got $z\cdot(x+y)=x\cdot y$

This is from "Test of math at 10+2 level": A vessel contains $x$ gallons of wine and another contains $y$ gallons of water. From each vessel $z$ gallons are taken out and transferred to the other. ...
2
votes
0answers
75 views

Proof for the distributivity of multiplication over addition for a Binary Field

For the standard binary field $\mathbb{F}_{2} = \{0, 1\}$. Where the operations of addition and multiplication exist, and multiplication is equivalent to logical and, and addition is equivalent to ...
3
votes
0answers
56 views

Recursive formula: Probability that a ball is drawn from a jar [DBertsekas P56, 1.19]

Each of $k$ jars contains $w$ white and $b$ black balls. A ball is randomly chosen from jar 1 and transferred to jar 2, then a ball is randomly chosen from jar 2 and transferred to jar 3, etc. ...
3
votes
1answer
43 views

B suggests A iff B Complement Doesn't Suggest A [DBertsekas P58 1.25(b)]

Let $A$ and $B$ be events with $P(A) > 0$ and $P(B) > 0$. We say that an event $B$ suggests an event $A$ if $P(A|B) > P(A)$, and does not suggest event A if $P(A|B) < P(A)$. ...
2
votes
2answers
72 views

Proving the derivative is $0$ at the extremum and all derivatives are $0$.

The pictures below show the proof that Apostol uses in his book. I can't understand why Apostol introduces the function $Q(x)$ and proves the theorem by contradiction using the sign preserving ...
3
votes
0answers
79 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
4
votes
3answers
591 views

Combinatorial Proof of Multinomial Theorem - Without Induction or Binomial Theorem

I've been trying to rout out an exclusively combinatorial proof of the Multinomial Theorem with bounteous details but only lighted upon this one - see P2. Any other helpful ones? ...
0
votes
1answer
54 views

Intuitive proof of interpolation polynomial existence

Problem: Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n ...
1
vote
2answers
513 views

Proof that negative binomial distribution is a distribution function?

In my textbook, a clear proof that the Geometric Distribution is a distribution function is given, namely $$\sum_{n=1}^{\infty} \Pr(X=n)=p\sum_{n=1}^{\infty} (1-p)^{n-1} = \frac{p}{1-(1-p))}=1.$$ ...
1
vote
2answers
327 views

Using inequalities and limits

Is it possible to say: $$ If \ f(x) \ and \ g(x) \ both \ have \ limits \ as \ x\to p\ and \ f(x) \le g(x), \ then \lim_{x \to p} f(x)\le \lim_{x \to p} g(x). $$ My proof(Edit: Proof is wrong due to ...
0
votes
1answer
72 views

Norm of a regular ideal of an order of an algebraic number field which is Galois over $\mathbb{Q}$

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank ...
9
votes
1answer
163 views

Is my proof correct for: $\sqrt[7]{7!} < \sqrt[8]{8!}$

I have to show that $$\sqrt[7]{7!} < \sqrt[8]{8!}$$ and I did the following steps \begin{align} \sqrt[7]{7!} &< \sqrt[8]{8!} \\ (7!)^{(1/7)} &< (8!)^{(1/8)} \\ (7!)^{(1/7)} - ...
0
votes
1answer
103 views

Decomposition of a primitive ideal of a quadratic order

Let $K$ be a quadratic number field. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because ...
0
votes
1answer
166 views

On the norm formula $N(IJ) = N(I)N(J)$ in an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module ...
2
votes
2answers
340 views

Proving the limits of the sum of two functions is equal to the sum of the limits

I am new to proving in math so I want to know if this informal proof of limits is possible: Theorem: If $\lim_{x \to a}f(x)=A$ and $\lim_{x \to a}g(x) = B$, then $$\lim_{x \to a}[f(x)+g(x)]=A+B$$ ...
0
votes
6answers
285 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
-2
votes
1answer
226 views

Criterion on whether a given ideal of a quadratic order is regular or not

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of rank ...
1
vote
2answers
1k views

Proving that any finite integral domain is a field.

Let $R$ be an integral domain, and let $a \in R, a \neq 0$. Let $f_a: R \rightarrow R$ be defined by $f_a(r)=ar$. Prove that $f$ is injective. Then prove that every finite integral domain is a ...
-2
votes
1answer
161 views

Decomposition of a primitive regular ideal of a quadratic order

Let $K$ be a quadratic number field. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because ...
0
votes
1answer
325 views

Norm of the product of two regular ideals of an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank ...
2
votes
1answer
110 views

Factor ring by a regular ideal of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is a finitely generated $A$-module. It is well-known that ...