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

0
votes
1answer
56 views

How to prove the External Bisector Theorem by dropping perpendiculars from a triangle's vertices?

I've found two different methods to prove Internal Angle Bisector Theorem, viz. Wikipedia ("Proof 2") method and AskMath.com method. How can we prove External Angle Bisector Theorem with ...
1
vote
1answer
36 views

Exercise about nbd-finiteness (Dugunji III.9.1)

Sorry for the vague title, but the question is fairly long: Let $\{A_\alpha\}$ be a ndb-finite closed cover of $X$. Consider $x_0\in X$, and let $A_{\lambda_i}$ be (all) the $A_\alpha$ that ...
0
votes
5answers
126 views

Different proofs that $\lim_{n\to\infty}\sin n$ does not exist [duplicate]

In this question it was proved that limit $$ \lim_{x\to\infty}\sin x $$ doesn't exists. What about $$ \lim_{n\to\infty}\sin n? $$ I asking about usual limit, where $n$ is integer. I know that this ...
0
votes
0answers
60 views

Easy proof of Cayley Hamilton theorem [duplicate]

What is wrong with this proof of Cayley hamilton? If $A$ is $n \times n$ matrix and $P$ is its characteristic polynomial then $P(A) = 0.$ Proof: $P(x) = \det(A - xI) \implies P(A) = \det(A - A) = ...
3
votes
4answers
254 views

On the value of proofs vs counterexamples

If a conjecture doesn't hold we usually provide a counterexample. While re-proving theorems is valuable and mathematicians do it usually, I think proving that some statement is wrong without giving ...
7
votes
4answers
214 views

Looking for a direct proof of the following exercise

A friend of mine told me about the following problem: Let $\{r_n\}$ be a sequence of rational numbers such that $\lim_{n\to\infty}r_n=x\in\Bbb R,$ $r_n\neq x,$ for every $n\in\Bbb N$ and ...
1
vote
2answers
58 views

What is your favorite proof of the Pythagorean Theorem? Why? [duplicate]

My favorite is Euclid's original proof for two reasons: First, it requires minimal raw material. It only needs the result that the area of a triangle is half the area of a rectangle with the same ...
3
votes
0answers
60 views

Are there less trivial necessary and sufficient conditions?

Given an infinite set $X$ with the finite-complement topology, find a necessary and sufficient condition for a map $f:X\to X$ to be continuous. I came up with the condition that $\lvert ...
2
votes
0answers
29 views

The proof of the integral test using the contradiction method.

I am currently writing a short note about the proof techniques. I found a random theorem and wanted to write a proof by contradiction as an example. The theorem says The integral ...
2
votes
0answers
99 views

Euler-Mascheroni constant [strategic proof]

I know two proof about the approximation of Euler-Mascheroni constant $\gamma$, but very technical. So I would like to know if someone has a strategic proof to show that $0,5<\gamma< 0,6.$ ...
0
votes
1answer
54 views

If $\{a_{n_k}\}$ is a subset of $\{a_n\}$, $\lim_{k\to\infty} a_{n_k }= \lim_{n\to\infty} a_n\ $

Let $\{a_n\}$ be a sequence and L a real number such that $\lim_{n\to\infty} a_n = L$ Prove that if $\{a_{n_k}\}$ is any subsequence of $\{a_n\}$, then $\lim_{k\to\infty} a_{n_k} = L $ I have ...
0
votes
1answer
36 views

Prove that $\{a_n\}$ is bounded [duplicate]

Let $\{a_n\}$ be a sequence an $L$ a real number such that $\lim_{n\to\infty} a_n = L$ Prove that $\{a_n\}$ is bounded This reminds me of the bounded monotone convergence theorem (BMCT) but in ...
20
votes
1answer
324 views

Does Nakayama Lemma imply Cayley-Hamilton Theorem?

Consider the Cayley-Hamilton Theorem in the following form: CH: Let $A$ be a commutative ring, $\mathfrak{a}$ an ideal of $A$, $M$ a finitely generated $A$-module, $\phi$ an $A$-module endomorphism ...
6
votes
2answers
135 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 ...
2
votes
0answers
46 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 ...
0
votes
0answers
59 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 ...
1
vote
1answer
46 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!} ...
0
votes
1answer
94 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
52 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
49 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
18 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 ...
1
vote
1answer
121 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 ...
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 ...
0
votes
1answer
52 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 ...
3
votes
3answers
95 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 ...
5
votes
4answers
420 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
95 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 $p$ ...
-1
votes
5answers
134 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
98 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
0answers
82 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 ...
0
votes
1answer
16 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 ...
4
votes
7answers
92 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, ...
8
votes
2answers
151 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. ...
2
votes
2answers
72 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 ...
2
votes
0answers
60 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) = ...
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
37 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 ...
53
votes
15answers
2k 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
59 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. ...
1
vote
1answer
65 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 ...
1
vote
2answers
33 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 ...
6
votes
1answer
118 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 ...
0
votes
0answers
48 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 ...
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
21 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 ...
3
votes
2answers
88 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 ...
2
votes
0answers
75 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 ...
1
vote
1answer
45 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 ...
32
votes
1answer
561 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 ...