Tagged Questions

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

learn more… | top users | synonyms

1
vote
0answers
37 views

Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
2
votes
2answers
324 views

How to complete this proof to show that the metric $d'(x,y) = d(x,y) / (1 + d(x,y))$ gives the same topology as $d(x,y)$ gives?

This is an exercise problem from Munkres's Topology (Exercise 11 of Section 20 "The Metric Topology", 2nd edition). Exercise 11: Show that if $d$ is a metric for $X$, then $$d'(x,y) = d(x,y) / (1 ...
1
vote
1answer
55 views

Proving via axioms, that for given set $A$, $P(P(A))$ exists

The question itself: For a given set A, prove P(P(A)) exists. You may only use the axiom of pairing, axiom of union and axiom of empty set. This is how I solved it: Let A be the given set. ...
10
votes
1answer
171 views

Direct combinatorial proof of a sum identity on formal Lagrange polynomials

Let $k$ be a field and $K=k(x_0,x_1,\ldots, x_n)[x]$. Define $$\mathcal{L}_k(x)\triangleq \prod_{\substack{j=0\\ j\ne k}}^n\frac{x-x_j}{x_k-x_j}.$$ Is there a purely combinatorial way to show ...
2
votes
2answers
129 views

The center of a group with order $p^2$ is not trivial

Let $p$ be a prime and $G$ be a group of order $p^2$. Show that $Z(G)\neq 1$. Is there a proof of this nice fact that doesn't use the class equation?
0
votes
0answers
96 views

The Toad and Frog Game - Proof by Inducation

Toads and Frogs is played on a 1 × n strip of squares. At any time, each square is either empty or occupied by a single toad or frog. Although the game may start at any configuration, it is customary ...
16
votes
2answers
948 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as briefly as possible

Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that $$ x^3 + y^3 = z^3, $$ has no positive integer solutions, as briefly as possible? I have seen some ...
3
votes
1answer
96 views

How to show that $\lfloor n/1\rfloor+\lfloor n/2 \rfloor+…+\lfloor n/n\rfloor+\lfloor{\sqrt{n}}\rfloor$ is even?

Let $n$ is a natural number. Prove that $$\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+....+\left\lfloor\frac{n}{n}\right\rfloor+\left\lfloor{\sqrt{n}}\right\rfloor$$ ...
2
votes
0answers
103 views

The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
0
votes
2answers
106 views

Matrix power series has linearly dependent terms

Prove that for any $(n\times n)$ real matrix, the set of matrices $\{I,M,M^2,...,M^n\}$ are linearly dependent. More formally, we have to prove that $$\forall M \in \mathbb{R}^{n \times n},\\ ...
1
vote
0answers
943 views

Easier Solution? - Find plane perpendicular to another plane and through the intersection line of two planes [Stewart P803 12.5.38]

$38.$ Find an equation of the plane that's $\perp$ the plane $x + y - 2z = 1$ and passes through the line of intersection of the planes $x - z = 1$ and $y + 2z = 3$. $\bbox[3px,border:2px solid ...
1
vote
1answer
265 views

Problem with alternate solution — Equation of plane through point and containing intersection line of two planes [Stewart P $803, 12.5.37$]

$37.$ Find an equation of the plane that passes through the point $(1, -2, 1)$ and contains the line of intersection of the planes $x + y - z = 2$ and $2x - y + 3z = 1$. $\bbox[3px,border:2px solid ...
1
vote
0answers
98 views

How to prove that the inverse of a persymmetric matrix is also persymmetric?

An exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...
3
votes
0answers
48 views

Using an automatic tool for checking geometric conjectures

I do a lot of research about squares, and I thought of using some automatic tool for proving / disproving some geometric conjectures. As a simple example, consider the following Square coloring ...
3
votes
1answer
180 views

An alternate analysis to the (worst-case) run time of the euclidean algorithm

I was trying to figure out the running time of the euclidean algorithm. The analysis that I found on Wikipedia and CLRS both analyze the run time of the euclidean algorithm using the Fibonacci ...
7
votes
1answer
383 views

Uniform Convergence verification for Sequence of functions - NBHM

Following is a list of problems from an exam for admission into Ph.D program. I have just compiled all previous questions on uniform convergence of sequence of functions and i tried to work out . I ...
0
votes
2answers
63 views

How do I show that these are the same logical statement?

I know that if I wanna show that the following statement are the same, I may use some rules in Logic: $$P\Longrightarrow Q,\quad [P \text{ and } (\sim Q)]\Longrightarrow [R\text{ and }\sim R]$$ Is ...
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
61 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
78 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
197 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
136 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
213 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
74 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
257 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
120 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
166 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
82 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
108 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
106 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
199 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
293 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
87 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
50 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
77 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
4answers
525 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
290 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
103 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
68 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
226 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
67 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
122 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
46 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
530 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
54 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
92 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
88 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
72 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 $P(A|B) < P(A)$. ...