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

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5
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2answers
118 views

Determine the number of subsets

How many distinct subsets of a set $\text{A}$ are there, containing at least $9$ elements, where the total number of elements in set $\text{A}$ is $18$ ? I've solved it by making cases of either ...
1
vote
0answers
42 views

A better way to answer this question

So my team and i were asked this question a few years ago on a small Math-A-Thon on my hometown. It went something like this: "We need to transport a neon tube (or any tube, who cares) of 92cm ...
1
vote
1answer
118 views

Change of variable formula for the image of a hypercube

Let $\varphi: \mathbb{R}^n\to \mathbb{R}^n$ be an injective $C^1$ map. Let $I=[0, 1]^n$. I want to show that $$m(\varphi(I))=\int_I \left|\det D\varphi(x)\right|dx.$$ This is a special case of the ...
5
votes
1answer
673 views

Characterization of positive definite matrix with principal minors

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, ...
0
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0answers
53 views

Continuous function $0$ on one closed set and $1$ on the other

Looking for a better approach of the following question if possible. Question: Let $A$ and $B$ be disjoint nonempty closed sets in a metric spaces $X$, and define ...
-3
votes
2answers
28 views

Prove Sum Approximation Theorem [closed]

Prove if $S=\sum_{n=0}^{\infty}a_{n}x^{n}$ converges for $|x|<1$, and if $|a_{n+1}|<|a_{n}|$ for $n>N$, then $$|S-\sum_{n=0}^{N}a_{n}x^{n}|<|a_{N+1}x^{N+1}|/(1-|x|)$$ I have already proven ...
1
vote
1answer
44 views

Searching simpler proof for convergence of a sequence

It is known that if $f_n \to f$ uniformly and $x_n \to x$ then $f_n(x_n) \to f(x)$. As an example, this can be applied in order to show that $$\sum_{k=0}^n \frac{\left( 1-\frac{1}{n} \right)^k}{k!} ...
48
votes
1answer
2k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
0
votes
1answer
78 views

Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem

Prove that every Hamel basis in an infinite dimensional Banach Space is uncountable without using Baire Category Theory. We are assuming axiom that vector space dimension (if exist) is well-defined. ...
2
votes
2answers
50 views

Alternative Quadratic Formula

Well the formula for solving a Quadratic equation is : $$\text{If }\space ax^2+bx+c=0$$ then $$x=\dfrac{-b \pm \sqrt{b^2 -4ac} }{2a}$$ But looking at this : [Wolfram Mathworld] (And also in other ...
1
vote
1answer
47 views

Prove that the image of a a closed and bounded interval in $\mathbb{R}$ is a a closed and bounded interval in $\mathbb{R}$?

According to the excercise 7.22 of the book Topology by Franzosa: Combining the Extreme Value Theorem and the Intermediate Value Theorem, prove the following theorem: Let $[a, b]$ be a ...
2
votes
3answers
17 views

Direct Proof for Statement on Linear Independence and Unique Representations

The Statement Show that if a set of vectors is linearly independent, then any vector in the span of that set has a unique representation as a linear combination of these vectors. My Proof I'm going ...
45
votes
13answers
1k views

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
1
vote
1answer
67 views

What is $\lim_{n \to \infty} n^3 a_n$? [duplicate]

$a_n$ is the Fourier coefficient of $$f(x) = \left(1 - \frac{|x|}{\pi}\right)^4$$ The answer is infinity, but can someone give an answer that doesn't require explicit computation of the $a_n$? I'm ...
5
votes
4answers
390 views

Is there any published research on the value of finding new proofs for old theorems?

There have been many conjectures in history of mathematics that some of them after passing long journey have resulted in lengthy and high-level-math proofs. Perelman's proof on the Poincare's ...
6
votes
2answers
92 views

Alternative way to count the number of solutions to the equation $x^2 + y^2 = -1$ over $\Bbb Z /p$

$x^2 + y^2 = -1$ is a weird equation because it has no solutions over $\Bbb R$. I want to count the number of solutions it has over $\Bbb Z / p$ where $p$ is prime. If $p = 2$ then it has $p$ ...
0
votes
1answer
47 views

Proving a differential inequality without performing iteration

I'm seeking a better proof of the following fact: If $g$ is a non-negative bounded function, $g(0)=0$ and $g'(t)\leq \sqrt{g(t)}$ for all $t>0$, then $g(t)\leq t^2/4$. The upper bound $t^2/4$ is ...
4
votes
0answers
74 views

$\operatorname{Ext}^n$: computation verification

I would like someone to verify my computation of $\operatorname{Ext}^n$. Problem: Let $p$ be a prime, $k$ a field of characteristic $p$, $G = \langle x \mid x^p = 1 \rangle$, $B = kG$, $S = k(1 ...
3
votes
3answers
83 views

How is $\sin 45^\circ=\frac{1}{\sqrt 2}$?

I've been reading about the proof of $\sin 45^\circ=\dfrac{1}{\sqrt 2}$ in my book. They did it as following, let $\triangle ABC$ be an isosceles triangle as shown, Since the triangle is isosceles ...
1
vote
1answer
23 views

Question about required rigour in mappings proof

I was just working on some intro problems from an algebra textbook, and one of the proofs I had seemed to make sense to me, but when I compared it to a solution given online, it was seemingly very ...
0
votes
1answer
890 views

Proving graph connectedness given the minimum degree of all vertices

I know that this is a repeat of a previous question asked with a similar title, but I didn't want to revive an old thread. The solution presented in that thread seems to be the common one, but I was ...
30
votes
10answers
1k 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 ...
2
votes
2answers
68 views

$\frac{1}{{9\choose r}} -\frac{1}{{10\choose r}} = \frac{11}{6{11\choose r}}$. Is there a way to find $r$ without using algebra?

$$\frac{1}{\dbinom 9r} -\frac{1}{{\dbinom{10}r}} = \frac{11}{6\times \dbinom{11}r}$$ I guess directly applying algebra for this problem would be enough. But are there any simpler and prettier ...
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votes
5answers
123 views

Limits without L'Hospital

Evaluate: $$\lim_{h \rightarrow 0} \frac{e^{2h}-1}{h}$$ Now one way would be using the Maclaurin expansion for $e^{2x}$ However, can we solve it using the definition of the derivative (perhaps ...
2
votes
6answers
92 views

Alternative proof of $\cos^6{\theta}+\sin^6{\theta}=\frac{1}{8}(5+3\cos{4\theta})$

In this question, the only proof of the trigonometric identity: $$\cos^6{\theta}+\sin^6{\theta}=\frac{1}{8}(5+3\cos{4\theta})$$ is via factoring the sum of cubes: ...
4
votes
7answers
91 views

Show an $\arctan$ and $\arcsin$ function is constant

Show that for every $x\geq1$ the following is true: $2\arctan x + \arcsin \frac{2x}{1+x^2} = \pi$ One way (mentioned in the link at the bottom) would be to calculate the derivative of the left side, ...
0
votes
1answer
13 views

Parallelpiped formula induction

Good one guys! I've been able to prove (a) and (b), but (c) just got me struggling for a week now, and when I asked my orientator for help he said that I had to prove the parallelpiped diagonal ...
7
votes
2answers
133 views

Proving $\int_{0}^{\pi/2}x\sqrt{\tan{x}}\log{\sin{x}}\,\mathrm dx=-\frac{\pi\sqrt{2}}{48}(\pi^2+12\pi \log{2}+24\log^2{2}) $

When trying to solve this problem: How to Integrate $ \int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$ I found his sister integral has an interesting closed form provided my calculation is correct. ...
4
votes
6answers
193 views

Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third ...
2
votes
0answers
52 views

Proof Verification for Putnam Problem [Alternate Solution] 1997 A4

I have come across an interesting problem from the Putnam 1997 test, question A4: Problem: Let $G$ be a group with identity $e$ and $\phi: G \to G$ a mapping such that $\phi(g_1)\phi(g_2)\phi(g_3) = ...
1
vote
1answer
55 views

Did I solve exercise 4.5.4 (b) of 'How to Prove it' by velleman correctly and concisely?

4.5.4 Suppose R is a strict partial order on A. Let S be the reflexive closure of R. (b) Show that if R is a strict total order, then S is a total order. Suppose R is a strict total order. ...
4
votes
2answers
41 views

Proving a formula using another formula

These questions are from the book "What is Mathematics": Prove formula 1: $$1 + 3^2 + \cdots + (2n+1)^2 = \frac{(n+1)(2n+1)(2n+3)}{3}$$ formula 2: $$1^3 + 3^3 + \cdots + (2n+1)^3 = ...
2
votes
0answers
29 views

Clever proofs that prove one identity is equal to another, without going through the original identity?

In a previous question I attempted to formalize the argument of going from one proof of an identity to another, which turned out to be harder than I thought. The thing is, while it may be impossible ...
1
vote
1answer
59 views

Finding all solutions to $x^2+y^2=2010$

I need to find all integer solutions to $x^2+y^2=2010$. we can take $x\leq y$ for commodity. The problem can be tackled through brute force. We need $1005\leq x^2\leq 2010$ and so $32\leq x \leq ...
0
votes
0answers
59 views

Finite solution of Power Diophantione Equation.

Given an equation $x^2+k=y^3$ where k is a constant and $y=f(x)$,$f(x)$ is differentiable and algebraic. for which- $$\frac{d}{dx}x^{2} \neq\frac{d}{dx} f(x)^3$$ 1. Can I infer that the ...
1
vote
3answers
492 views

Proof of the duality of the dominance order on partitions

Could anyone provide me with a nice proof that the dominance order $\leq$ on partitions of an integer $n$ satisfies the following: if $\lambda, \tau$ are 2 partitions of $n$, then $\lambda \leq \tau ...
6
votes
1answer
116 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|].$$ This question is a re-posting of An expectation inequality. I can ...
1
vote
2answers
28 views

Extreme value theorem, without Heine Borel.

I was wondering, if there are any mistakes, in this proof of the extreme value theorem: Theorem. Let $X$ be a compact set and $f:X\rightarrow\mathbb{R}$, s.t. $f$ is continuous. Then there exists ...
1
vote
1answer
41 views

A question on Primes in Arithmetic Progression

We know that an arithmetic progression has to have a composite number since there are arbitrarily large gaps between primes. But I was wondering whether the following construction is possible: Can ...
2
votes
2answers
140 views

Proof for $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ without complexes? [duplicate]

This is what I needed. Practically, a link were also okay. $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$
29
votes
1answer
518 views

Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In a gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality ...
0
votes
0answers
35 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
5
votes
4answers
5k views

Why Markov matrices always have 1 as an eigenvalue

Also called stochastic matrix. Let $A=[a_{ij}]$ - matrix over $\mathbb{R}$ $0\le a_{ij} \le 1 \forall i,j$ $\sum_{j}a_{ij}=1 \forall i$ i.e the sum along each column of $A$ is 1. I ...
1
vote
1answer
492 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 ...
0
votes
0answers
24 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
16 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 ...
19
votes
6answers
686 views

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...
2
votes
0answers
67 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 ...
3
votes
2answers
76 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
63 views

Question 7F from general topology by Stephen willard?

Can someone help me with 7F from Willard? In part two : $\mathbf{7}$F. Functions to and from the plane. The facts presented here for the plane will be proved in more generality for ...