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

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3
votes
1answer
77 views

Solving the trigonometric equation $\tan^2x+\cot^2x=2-\cos^{2014}(2x)$

I was solving the trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. $$ L.H.S=\tan^2x+\cot^2x =\tan^2x+\frac{1}{\tan^2x} ...
1
vote
0answers
20 views

Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
0
votes
2answers
42 views

Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$.

I have been working through the following proof: Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$. Below, I have included screenshots of the ...
0
votes
3answers
58 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
3
votes
4answers
97 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...
2
votes
2answers
44 views

What exactly is $\cap$-stable here?

From my lecture notes: Theorem: Let $(\Omega, \mathcal A, P)$ be a probability space, $A \in \mathcal A, \mathcal M := \{ M_1, \ldots, M_n \} \subset \mathcal A$. The following statements are ...
0
votes
2answers
54 views

Prove that if $a<1/a<b<1/b$ then $a<-1$

The following is Exercise 3.2.8 from Velleman: Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<1/a<b<1/b$ then $a<-1$. I solved it using the hint in the back of ...
1
vote
2answers
14 views

A finite set and the set of its fixed points under any involution have cardinalities of the same parity

I am trying to write down a formal proof of the following fact: Let $A$ be a non-empty finite set and $f$ an involution on $A$. If $A'$ is the set of fixed points of the involution $f$, then $|A| ...
2
votes
3answers
61 views

Prove that $\sum_{d|n}\phi(d)=n$ where $\phi$ is the Euler's phi function, $n,c\in\mathbb{N}$

Here is a very elementary number theory proof using strong induction. Please mark/grade. Prove that $$\sum_{d|n}\phi(d)=n$$where $\phi$ is the Euler's phi function, $n,d\in\mathbb{N}$ First, ...
1
vote
1answer
23 views

Asymptote criterion

Let $f:(a, \infty)\to \Bbb R$ be a differentiable function such that exists $\lim_{x\to\infty}f(x)=l<\infty$ and exists (in the sense it can also be infinity) $\lim_{x\to\infty}f'(x)$. Under these ...
0
votes
1answer
38 views

Prove that $H\times K \cong K\times H$

According to the book: Let $G$ be the internal direct product of subgroups $H$ and $K$. Then $G$ is isomorphic to $H\times K$. From that it results $H\times K \cong K\times H$. Is there any ...
8
votes
2answers
122 views

Why is periodic harmonic analysis only possible with sines?

This paper shows that if we consider odd functions on $(-\pi,\pi)$ in $L_2$, then the only $2\pi$-periodic function $f$ for which $f(nx)$ is a complete orthogonal system is the sine function. I'll ...
0
votes
0answers
26 views

Is my proof of 'inscribed angle theorem' different from the usual one?

The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is ...
3
votes
3answers
167 views

Possible alternative way of expressing continuity of a function?

In Calculus or Real Analysis the usual form of definition of continuity of a function is $\epsilon- \delta$ def. From a rigorous point of view, is it possible to say this way? and if so, why?: ...
2
votes
2answers
34 views

Expected value of cube projection

This problem is from the book of V. I. Arnold. Find the expected area of the projection of a unit cube onto the plane under isotropic random direction of projection. The direct evaluation of it ...
2
votes
0answers
49 views

proof about triangles with sides in arithmetic progression

$\require{cancel}$ The sides of a triangle are in an arithmetic progression with $k>0$ and $a>b>c$. Prove that ...
0
votes
1answer
48 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
35 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
124 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
58 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
250 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
212 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
52 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
58 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
27 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
58 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
53 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
35 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 ...
19
votes
1answer
286 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 ...
5
votes
2answers
120 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
45 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
56 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
45 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
84 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
48 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 ...
1
vote
1answer
120 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
88 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
407 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
94 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
132 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
97 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
81 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
15 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
142 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. ...