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
1answer
71 views

Proof without mean value theorem

Is it possible to prove the following without using the mean value theorem: If $f$ is differentiable on an interval containing $0$ and if $\lim_{x \to 0} f'(x) = L$ then $f'(0) = L$. I have ...
13
votes
1answer
264 views

A beautiful inequality for convex functions

Let $f\in \mathcal{C}([0,1],\mathbb R_+)$ increasing. Prove that there exist $g,h\in \mathcal{C}([0,1],\mathbb R)$, convexs, such that $g\leqslant f \leqslant h$ and : $$\displaystyle ...
17
votes
8answers
425 views

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
0
votes
2answers
55 views

Consecutive positive integers proof problem

Consider any three consecutive positive integers. Prove that the cube of the largest cannot be the sum of the cubes of the other two. Work: I tried to prove via contradiction. I made three ...
1
vote
0answers
28 views

Dimension Theorem modification

The Dimension Theorem says $$ \dim(U+W) = \dim(U) + \dim(W) - \dim(U \cap W) $$ The proof of this theorem uses the bases of $U$, $W$, and $U\cap W$. Is it possible to prove this theorem with just ...
2
votes
2answers
99 views

How to prove from the definition that $X_n \xrightarrow{\mathbb{P}} X$ implies $\frac{1}{X_n} \xrightarrow{\mathbb{P}} \frac{1}{X}$?

Let $X, X_1, X_2, \ldots : \Omega \to (0,\infty)$ be random variables such that $X_n \xrightarrow{\mathbb{P}} X$. I'd like to show from the definition that $\frac{1}{X_n} \xrightarrow{\mathbb{P}} ...
10
votes
4answers
289 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know we can use Weierstrass theorem but I'd like to ...
0
votes
0answers
50 views

Extreme Value Theorem: from 1 variable to several variables?

When studying functions of several variables, one useful idea is to use the same result for one variable. For example, to prove that a local extremum of a function $f(x_1,\dots,x_n)$ is a critical ...
2
votes
0answers
93 views

Back-and-Forth Argument vs. “One-Way” Argument

The wikipedia article on the Back and Forth Argument claims at the end: If we iterated only step $(1)$, rather than going back and forth, then in some cases the resulting function from A to B ...
1
vote
1answer
32 views

Compact embedding

Prove that the embedding $j\colon (C^1[0,1],\|\cdot\|)\to(L^1[0,1],\|\cdot\|_{L^1})$ where $\|f\|=\max\{\|f\|_\infty,\|f'\|_\infty\}$ and $\|f\|_\infty$ denotes the supremum norm, ...
1
vote
0answers
34 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
180 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
52 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
159 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
122 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
52 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
760 views

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

Can someone prove 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 simply as possible? I have seen some good proofs, but ...
2
votes
1answer
73 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
82 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
84 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
272 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
149 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
56 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
43 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
126 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 ...
6
votes
1answer
259 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
62 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
50 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
55 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
70 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
144 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
4answers
191 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
209 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
115 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
161 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
98 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
32 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
99 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
159 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
83 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
44 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
74 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
391 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
189 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
93 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
0answers
56 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. ...